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Dr.  Horace  Ivie 


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ECLECTIC    EDUCATIONAL    SERIES. 


BATS   NEW  HIGHER   ALGEBRA. 


ELEMENTS 


OF 


ALGEBRA, 


FOR  ^  .  ^ 


COLLEGES,  SCHOOLS,  AND  PRIVATE  STUDENTS. 


By  JOSEPH  RAY,  M.  D., 

LATE    PROFESSOR  OF   MATHEMATICS   IN    WOODWARD  COLLEGE. 


Edited  by  DEL.  KEMPER,  A.  M.,  Prof,  op  Mathematics,  Hamden  Sidney 
College. 


NEW-YORK     •:.     CINCINNATI     •:.     CHICAGO 

AMERICAN    BOOK    COMPANY 

FROM  THE  PBE88  OF 
VAN  ANTWERP,  BRAGG,  A,  CO. 


Ray's  Mathematical  Series. 


^  S^t^^ 


111)  r,    \\  6  )r  «uc^  Hlv  \tZv\  O   -^ 
ARITHMETIC.  FVL     ^ 

Ray's  New  Primary  Arithmetic. 
Ray's  New  Intellectual  Arithmetic. 
Ray's  New  Practical  Arithmetic. 
Ray's  New  Higher  Arithmetic. 

TWO-BOOK   SERIES. 

Ray's  New  Rlementary  Arithmetic. 
Ray's  kW  Practical  Arithmetic. 

AI.GEBRA. 

Ray's  New  Elementary  Algebra. 
Ray's  New  Higher  Algebra. 

HIGHER  MATHEMATICS. 

Ray's  Plane  and  Solid  Geometry. 

Ray's  Geometry  and  Trigonometry. 

Ray's  Analytic  Geometry. 

Ray's  Elements  of  Astronomy. 

Ray's  Surveying  and  Navigation. 

Ray's  Differential  and  Integral  Calculus. 

eour.ATJQN  pei^ 

Entered  according   to  Act  of  Congress,  in  the  year    1852,  by  W,  B.  Smith,  in   the 
Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the  District  of  Ohio. 

Entered   according   to  Act  of  Congress,  in    the  year   1866,  bv  Sabgent,  Wilson  & 

HiNKLE,  in  the  Clerlj's  Office  of  the  District  Court  o'f  the  United 

States  for  the  Southern  District  of  Ohio. 


PREFACE, 


Algebra  is  justly  regarded  one  of  the  most  interesting  and 
useful  branches  of  education,  and  an  acquaintance  with  it  is  now 
sought  by  all  who  advance  beyond  the  more  common  elements. 
To  those  who  would  know  Mathematics,  a  knowledge  not  merely 
of  its  elementary  principles,  but  also  of  its  higher  parts,  is  essen-- 
tial;  while  no  one  can  lay  claim  to  that  discipline  of  mind  which 
education  confers,  who  is  not  familiar  with  the  logic  of  Algebra. 

It  is  both  a  demonstrative  and  a  practical  science— a  system 
of  truths  and  reasoning,  from  which  is  derived  a  collection  of 
Kulee  that  may  be  used  in  the  solution  of  an  endless  variety  of 
problems,  not  only  interesting  to  the  student,  but  many  of  which 
are  of  the  highest  possible  utility  in  the  arts  of  life. 

The  object  of  the  present  treatise  is  to  present  an  outline  of 
this  science  in  a  brief,  clear,  and  practical  form.  The  aim 
throughout  has  been  to  demonstrate  every  principle,  and  to  fur- 
nish the  student  the  means  of  understanding  clearly  the  rationale 
of  every  process  he  is  required  to  perform.  No  effort  has  been 
made  to  simplify  subjects  by  omitting  that  which  is  difficult,  but 
rather  to  present  them  in  such  a  light  as  to  render  their  acquisi- 
tion within  the  reach  of  all  who  will  take  the  pains  to  study. 

To  fix  the  principles  in  the  mind  of  the  student,  and  to  show 
their  bearing  and  utility,  great  attention  has  been  paid  to  the 
preparation  of  practical  exercises.  These  are  intended  rather  to 
illustrate  the  principles  of  the  science,  than  as  difficult  problems 
to  torture  the  ingenuity  of  the  learner,  or  amuse  the  already 
skillful  Algebraist. 

An  effort  has  been  made  throughout  the  work  to  observe  a 
natural  and  strictly  logical  connection  between  the  different 
parts,  so  that  the  learner  may  not  be  required  to  rely  on  a  prin- 

924223       (") 


iv  PREFACE. 

ciple,  or  employ  a  process,  with  the  rationale  of  which  he  is  not 
already  acquainted.  The  reference  by  Articles  will  always  en- 
able him  to  trace  any  subject  back  to  its  first  principles. 

The  limits  of  a  preface  will  not  permit  a  statement  of  the 
peculiarities  of  the  work,  nor  is  it  necessary,  as  those  who  are 
interested  to  know  will  examine  it  for  themselves.  It  is, 
however,  proper  to  remark,  that  Quadratic  Equations  have 
received  more  than  usual  attention.  The  same  may  be  said  of 
lladicals,  of  the  Binomial  Theorem,  and  of  Logarithms,  all  of 
which  are  so  useful  in  other  branches  of  Mathematics. 

On  some  subjects  it  was  necessary  to  be  brief,  to  bring  the 
work  within  suitable  limits.  For  example,  what  is  here  given 
of  the  Theory  of  Equations,  is  to  be  regarded  merely  as  an 
outline  of  the  more  practical  and  interesting  parts  of  the  subject, 
which  alone  is  sufficient  for  a  distinct  treatise,  as  may  be  seen 
by  reference  to  the  works  of  Young  or  Hymers  in  English,  or 
of  DeFourcy  or  Reynaud  in  French. 

Some  topics  and  exercises,  deemed  both  useful  and  interesting, 
will  be  found  here,  not  hitherto  presented  to  the  notice  of  stu- 
dents. But  these,  as  well  as  the  general  manner  of  treating  the 
subject,  are  submitted,  with  deference,  to  the  intelligent  educa- 
tional public,  to  whom  the  author  is  already  greatly  indebted  for 
the  favor  with  which  his  previous  works  have  been  received. 

Woodward  College,  May,  1852. 

Publishers'  Notice. — This  work,  originally  published  as 
Ray's  Algebra,  Part  II.,  w^as  revi.sed,  in  1867,  by  Dr.  L.  D. 
Potter.  Portions  of  the  work  were  revised  in  1875,  by  Prof. 
Del.  Kemper. 


CONTENTS 


I.— FUNDAMENTAL  RULES. 

ARTICLES. 

Definitions  and  Notation 1 —  36 

Exercises  on  the  Definitions  and  Notation 36 

Examples  to  be  written  in  Algebraic  Symbols 36 

Addition — General  Rule. — Subtraction — Rule 37 —  45 

Bracket,  or  Vinculum 46 

Observations  on  Addition  and  Subtraction 47 —  61 

Multiplication — Preliminary  principle 52 —  53 

Rule  of  Coefficients — of  Exponents 55 —  56 

Rule  of  the  Signs— General  Rule 60—  61 

Multiplication  by  Detached  Coefficients  ....  62 

Remarks  on  Algebraic  Multiplication     ....  63 —  66 

Division — Rule  of  Signs — Coefficients — Exponents  ....  67 —  70 

Division  of  a  Monomial  by  a  Monomial ....  71 

Division  of  Polynomials  by  Monomials  ....  74 

Division  of  one  Polynomial  by  another  ....  75 —  76 

Division  by  Detached  Coefficients 77 


II.— THEOREMS,  FACTORING,  Etc. 

Algebraic  Thkorems — Square  of  the  sum  of  two  quantities  .  78 

Square  of  the  difference  of  two  quantities    ...  79 

Product  of  Sum  and  Difference 80 

Transfer  of  factors  in  a  fraction 81 

Any  quantity  whose  exponent  is  0  is  =1     ...  82 

a"»— 6"*  is  divisible  by  « — b 83 

a2"i_j2r«  is  divisible  by  a-f  6 85 

a2™+i-f62m+i  ig  divisible  by  a+6 86 

Factoring — Of  Numbers — Of  Algebraic  quantities  ....  87 —  95 

Greatest  Common  Divisor 96 — 108 

liKAST  Common  Multiple 109 — 113 

(v) 


vi  CONTENTS. 


III.— ALGEBEAIC  FEACTIONS. 

ARTICLES. 

Definitions — Proposition — Lowest  terms 114 — 119 

To  reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity   ...  121 

To  reduce  a  Mixed  Quantity  to  the  form  of  a  Fraction      .    .  122 

Signs  of  Fractions 123 

To  reduce  Fractions  to  a  Common  Denominator 125 — 126 

To  reduce  a  Quantity  to  a  Fraction  with  a  given  Denominator  127 — 128 

Addition  and  Subtraction  of  Fractions 129 — 130 

Multiplication  and  Division  of  Fractions 131 — 132 

Eeduction  of  Complex  Fractions 133 

Eesolution  of  Fractions  into  Series 134 

Miscellaneous  Propositions  in  Fractions 135 — 137 

Theorems  in  Fractions — Miscellaneous  exercises 138 — 139 


IV.— SIMPLE  EQUATIONS. 

Definitions  and  Elementary  principles 140 — 149 

Transposition — Clearing  of  Fractions 150 — 151 

Solution  of  Simple  Equations — Eule 152 — 153 

Questions  involving  Simple  Equations 154 

Simple  Equations  with  two  unknown  quantities 155 

Elimination  by  Substitution — Comparison — Addition,  etc.    .  156 — 158 
Problems  producing  Simple  Equations  containing  two  un- 
known quantities 159 

Simple  Equations  involving  three  or  more  unknown  quan- 
tities   160 

Problems  producing  Simple  Equations  containing  three  or 

more  unknown  quantities 161 


Y.— SUPPLEMENT  TO  SIMPLE  EQUATIONS. 

Generalization — Formation  of  Eules — Examples  ....  162 — 163 
Negative  Solutions — Discussion  of  Problems — Couriers  .  .  164 — 166 
Cases  of  Indetermination  and  Impossible  Problems  .  .  .  167 — 169 
A  Simple  Equation  bos  but  One  Eoot 170 


CONTENTS.      *  vii 


VI.— FORMATION    OF    POWERS— EXTRACTION     OF    ROOTS- 
RADICALS— INEQUALITIES. 

ARTICLES. 

Involution  or  Formation  of  Powers — Newton's  Method    .     .  172 

Square  Root  of  Numbers — Of  Fractions — Theorem      .     .     .  173 — 179 

Approximate  Square  Roots 180 

Square  Root  of  Monomials — Of  Polynomials 182 — 184 

Cube  Root  of  Numbers — Approximate  Cube  Roots  ....  185 — 189 

Cube  Root  of  Monomials— Of  Polynomials 190—191 

Fourth  Root— Sixth  Root— Nth  Root,  etc 192 

Signs  of  the  Roots-Nth  Root  of  Monomials 193—194 

Radical  Quantities — Definitions — Reduction  of  Radicals     .  195 — 203 

Addition  and  Subtraction  of  Radicals 204 

Multiplication  and  Division  of  Radicals 205* 

To  render  Rational  the  Denominator  of  a  Fraction  ....  206 

Powers  and  Roots  of  Radicals — Imaginary  Quantities  ,     .     .  207 — 210 

Theory  of  Fractional  Exj)onents 211 

Multiplication  and  Division  in  Fractional  Exponents  .     .     .  212 — 213 
Powers  and  Roots  of  Quantities  with  Fractional  Exponents  .  214 

Simple  Equations  containing  Radicals 216 

Inequalities — Propositions  I  to  V — Examples 217 — 2^:4 


VII.— QUADRATIC  EQUATIONS. 

Definitions — Pure  Quadratic  Equations — Problems    ....  224 — 229 

Affected  Quadratic  Equations 230 

Completing  the  Square — General  Rule — Hindoo  Method  ,     .  2:]1 — 232 

Problems  producing  Affected  Equations 233 

Discussion  of  General  Equation — Problem  of  Lights     .     .     .  234 — 239 

Trinomial  Equations — Definitions 240 

Binomial  Surds — Theorems — Square  Root  of  A±|/B  ...  241 

Varieties  of  Trinomials — Form  of  Fourth  Degree      ....  242 — 243 

Simultaneous  Quadratic  Equations 244 

Pure  Equations — Affected  Equations 245 — 250 

Questions  producing  Simultaneous  Quadratic  Equations    .     .  251 

Formulse — General  Solutions 262 

Special  Artifices  and  Examples 253 


viii  CONTENTS. 


VIII.— RATIO— PROPORTION— PROGRESSIONS. 

ARTICLES. 

Ratio — Kinds — Antecedent  and  Consequent 254 — 256 

Ratio — Multiplication  and  Division  of 259 

Ratio  of  Equality — Of  greater  and  less  Inequality     ....  260 

Ratio— Compound — Duplicate — Triplicate    .......  261 

Ratios — Comparison  of 262 

Proportion — Definitions 263 — 266 

Product  of  Means  equal  to  Product  of  Extremes 267 

Proportion  from  two  Equal  Products 268 

Product  of  the  Extremes  equal  to  the  Square  of  the  Mean     .  269 

Proportion  by  Alternation — By  Inversion 270 — 271 

Proportion  from  equality  of  Antecedents  and  Consequents    .  272 

Proportion  by  Composition — By  Division 273 — 274 

Proportion  by  Composition  and  Division 275 

Like  Powers  or  Roots  of  Proportionals  are  in  Proportion  .    .  276 

Products  of  Proportionals  are  in  Proportion 277 

Continued  Proportion — Exercises — Problems 278—279 

IIarmonical  Proportion 280 

Variation — Propositions — Exercises 281 — 290 

Arithmetical  Progression — Increasing  and  Decreasing  .    .  291 

Last  Term— Rule— Sum  of  Series— Rule— Table 292—294 

To  insert  m  Arithmetical  Means  between  two  numbers,  Ex.  16  294 

Geometrical  Progression — Increasing  and  Decreasing    .     .  295 

Last  Term— How  to  find  it— Sum  of  Series— Rule    ....  296—297 

Sura  of  Decreasing  Infinite  Geometrical  Series 299 

Table  of  General  Formulae 300 

To  find  a  Geometric  Mean  between  two  numbers,  Ex.  18  .     .  300 

To  insert  m  Geometrical  Means  between  two  numbers,  Ex.  19  300 

Circulating  Decimals — To  find  the  value  of 301 

IIarmonical  Progression — Proposition 302 — 303 

Problems  in  Arithmetical  and  Geometrical  Progression    .    .  304 

IX.— PERMUTATIONS— COMBINATIONS— BINOMIAL 
THEOREM. 

Permutations 305 — 307 

Combinations 308 — 309 


CONTENTS.  ix 


ARTU'LFS. 


Binomial  Theorem  when  the  Exponent  is  a  Positive  Integer  :^I0 

Binomial  Theorem  applied  to  Polynomials  .......  311 

X.— INDETERMINATE  COEFFICIENTS— BINOMIAL  THEOREM 
—GENERAL  DEMONSTRATION— SUMMATION  AND 
INTERPOLATION  OF  SERIES.   . 

Indeterminate  Coefficients — Theorem — Evolution       .     .     .  314 — 317 

Decomposition  of  Rational  Fractions 318 

IkNOMiAL  Theorem  for  any  Exponent — Application  of  .     .     .  319 — 321 

Extraction  of  Roots  by  the  Binomial  Theorem 322 

Limit  of  Error  in  a  Converging  Series 323 

Differential  Method  op  Series — Orders  of  Differences  .    .  324 

To  find  the  nth  term  of  a  Series — The  sum  of  n  terms  .     .     .  326 — 327 

Piling  of  Cannon  Balls  and  Shells 328 — 331 

Interpolation  of  Series 333 — 335 

Summation  of  Infinite  Series 336 — 338 

Recurring  Series 339—343 

Reversion  of  Series 344 — 346 


XI.-CONTINUEDFRACTIONS— LOGARITHMS— EXPONENTIAL 
EQUATIONS— INTEREST  AND  ANNUITIES. 

Continued  Fractions 347 — 356 

Logarithms — Definitions — Characteristic  Table 357 — 359 

Properties  of  Logarithms — Multiplication — Division     .     .     .  360 — 361 

Formation  of  Powers — Extraction  of  Roots 362 — 363 

Logarithms  of  Decimals— Of  Base— Of  0 364—368 

Computation  of  Logarithms — Logarithmic  Series      ....  370—373 

Naperian  Logarithms — Computation  of 375 

Common  Logarithms — Computation  of  by  Series       ....  377 

Single  Position 380 

Double  Position 381 

Exponential  Equations       .     .     .  ' 382 — 383 

Interest  and  Annuities — Simj)le  Interest 384 — 385 

Compound  Interest — Increase  of  Population 386 — 387 

Compound  Discount — Formulee — Annuities 388 — 391 


X  CONTENDS. 

XIL— GENERAL  THEORY  OF  EQUATIONS. 

ARTICLES. 

Definitions — General  Form  of  Equations 393 — 394 

An  Equation  whose  Root  is  a  is  divisible  by  x — a 395 

An  Equation  of  the  nth  Degree  has  n  and  only  n  roots      .     .  396 — 397 
Relations  of  the  Roots  and  Coefficients  of  an  Equation      .    .  398 

What  Equations  have  no  Fractional  Roots 399 

To  change  the  Signs  of  the  Roots  of  an  Equation    ....  400 

Number  of  Imaginary  Roots  of  an  Equation  must  be  even  .  401 

Descartes'  Rule  of  the  Signs 402 

Limits  of  a  Root — A  method  of  finding 403 

Transformation  of  Equations .     .  404 — 408 

Synthetic  Division — Transformation  of  Equations  by.     .     .  409 — 410 
Derived  Polynomials — Law  of — Transformation  by  ....  411 — 413 

Equal  Roots 414 

Limits  of  the  Roots  of  Equations 415 

Limit  of  the  greatest  Root— Of  the  Negative  Roots  ....  416—418 
Sturm's  Theorem 420 — 427 

XIII.— RESOLUTION  OF  NUMERICAL  EQUATIONS. 

Rational  Roots — Rule  for  finding 429 

Horner's  Method  op  Approximation 430 — 434 

Approximation  by  Double  Position 436 

Newton's  Method  op  Approximation 437 

Cardan's  Rule  for  Solving  Cubic  Equations 438 — 441 

Reciprocal  or  Recurring  Equations 442 

Binomial  Equations 443 — 444 


HIGHER  ALGEBRA. 


I.  DEFINITIONS. 

Article  1.  Mathematics  is  the  science  of  the  exaot 
relations  of  Quantity  as  to  Magnitude  or  Form. 

S.  duantity,  as  the  subject  of  mathematical  investiga- 
tions, is  any  thing  capable  of  being  measured,  or  about 
which  the  question  How  much?  may  be  asked.  It  may  be, 
1.  Geometric,  involving  Form ;  2.  Number. 

3.  Number  is  quantity  considered  as  composed  of  equal 
parts  of  the  same  kind,  each  called  the  unit;  and  the  magni- 
tude of  the  quantity  is  indicated  by  its  ratio  to  the  unit. 

4.  Numbers  are  represented  by  conventional  symbols. 
When  the  symbols  used  arc  general,  as  distinguished  from 
the  arithmetical  symbols,  viz.,  the  Arabic  numerals,  the 
process  of  investigation  is  called  Algebraic.  Hence,  we 
have  the  following  definitions : 

5.  Algebra  is  the  method  of  investigating  the  relations 
of  numbers  by  means  of  general  symbols. 

Remark. — It  should  be  remembered  that  the  word  ^^ quantiti/,^^ 
whenever  used  ia  algebra,  is  synonymous  with  '^'•numhery 

G.  The  algebraic  symbols  are  of  two  kinds:  1.  Symbols 
of  numbers;    2.  Symbols  of  relation. 

Numbers  are  usually  represented  by  letters  ;  as,  a,  6,  x,  y : 
sometimes,  of  course,  when  known,  by  the  Arabic  numerals. 

T«  The  symbols  of  relation,  usually  called  Signs,  are  the 
representatives  of  certain  phrases,  and  are  used  to  express 
operations  with  precision  and  brevity.  The  principal  alge- 
braic signs  are  :  =  -|-  —  X   -^  \^' 

11 


12  RAY  S  ALGEBRA,  SECOND  BOOK. 

8.  The  Sign  of  Equality,  =,  is  read  equal  to.  It  de- 
notes that  the  quantities  between  which  it  is  placed  are 
equal.  Thus,  £C=5,  denotes  that  the  quantity  represented 
by  x.  equals  5,<.    ^  ;.• ' 

■^;^,\.!]?hefSij5n  .({f  A4^i^  +,  is  read  pZ?fs.  It  denotes 
thai  the  quantity  to  which  it  is  prefixed  is  to  be  added. 
Thus,  a-\-h  denotes  that  h  is  to  be  added  to  a. 

10.  The  Sign  of  Subtraction,  — ,  is  read  minm.  It 
denotes  that  the  quantity  to  which  it  is  prefixed  is  to  be 
subtracted.  Thus,  a — h  denotes  that  h  is  to  be  subtracted 
from  a. 

11.  The  signs  -\-  and  —  are  called  (lie  signs.  The 
former  is  called  the  lyositioe^  the  latter  the  negative  sign  ; 
they  are  said  to  be  contrary^  or  opposite. 

12.  Every  quantity  is  supposed  to  have  either  the  posi- 
tive or  negative  sign.  When  a  quantity  has  no  sign  pre- 
fixed to  it,  -f-  is  understood.     Thus,  a=-\-a. 

Quantities  having  the  positive  sign  are  called  positive; 
those  having  the  negative  sign,  negative. 

13.  Quantities  having  the  same  sign  are  said  to  have 
like  signs  ;  those  having  different  signs,  unlike  signs. 

Thus,  -\-a  and  -{-&,  or  — a  and  — 6,  have  like  signs  ; 
while  +c  and  — d  have  unlike  sio;ns. 


o 


14.  The  Sign  of  Multiplication,  X,  is  read  into.,  or 
multiplied  hij.  It  denotes  that  the  quantities  between 
■which  it  is  placed  are  to  be  multiplied  together. 

The  product  of  two  or  more  letters  is  also  expressed  by 
a  dot  or  period,  or  by  writing  the  letters  in  close  succes- 
sion.    The  last  method  is  generally  to  be  preferred. 

Thus,  the  continued  product  of  the  numbers  designated 
by  a,  6,  and  c,  is  denoted  by  ayjjy^c.,  or  a.hx^  or  abc. 


DEFINITIONS.  13 

15.  Factors  are  quantities  that  are  to  be  multiplied 
together.  Thus,  in  the  product  a6,  there  are  two  fac- 
tors, a  and  h]  in  the  product  3x5X7,  there  are  three 
factors,  3,  5,  and  7. 

16,  The  Sign  of  Division,  -r-,  is  read  divided  hy.  It 
denotes  that  the  quantity  preceding  it  is  to  be  divided  by 
that  following  it. 

Division  is  also  expressed  by  placing  the  dividend  as  the 
numerator,  and  the  divisor  as  the  denominator  of  a  frac- 
tion. 

Thus,  a-^?>,  or  -,  signifies  that  a  is  to  be  divided  by  h. 

IT.  The  Sign  of  Inequality,  >,  denotes  that  one  of 
the  two  quantities  between  which  it  is  placed  is  greater 
than  the  other.  The  opening  of  the  sign  is  toward  the 
greater  quantity. 

Thus,  a^6,  denotes  that  a  is  greater  than  h.  It  is 
read  a  greater  than  h.  Also,  c<Cid,  denotes  that  c  is  less 
than  d,  and  is  read,  c  less  than  d. 

18.  A  Coefficient  is  a  number  or  letter  prefixed  to  a 
quantity,  to  show  how  many  times  it  is  taken. 

Thus,  if  a  is  to  be  taken  4  times,  instead  of  writing 
a-^a-\-a-\-a,  write  4a ;  also,  az-\-az-\-az=Saz. 

The  coefficient  is  called  numeral  or  literal^  according  as 
it  is  a  number  or  a  letter.  Thus,  in  the  quantities  hx  and 
wx,  5  is  a  numeral  and  m  a  literal  coefiicient. 

In  3«2,  3  may  be  considered  as  the  coefficient  of  az, 
or  3a  as  the  coefficient  of  z. 

When  no  numeral  coefficient  is  expressed,  1  is  understood. 
Thus,  a  is  the  same  as  la,  and  ax  the  same  as  \ax. 

lO.  A  Power  of  a  quantity  is  the  product  arising  from 
multiplying  the  quantity  by  itself  one  or  more  times. 

When  the  quantity  is  taken  twice  as  a  factor,  the  product 
is  called  the  second  power ;  when  three  times,  the  third 
power ;  and  so  ou. 


14  RAY'S  ALGEBRA,  SECOND  BOOK. 

Thus,  2x2=  4,  is  the  second  power  of  2. 

2X2X2=  8,  is  the  third  power  of  2. 
2X2X2X2=16,  is  the  fourth  power  of  2. 
Also,  aX^=   ^^)  is  t^he  second  power  of  a. 

ciy^ay^azzzaaa,  is  the  third  power  of  a;  and  so  on. 

The  second  power  is  often  termed  the  square,  and  the 
third  power,  the  cube. 

An  Exponent  is  a  small  figure  or  letter  placed  on  the 
right,  and  a  little  above  a  quantity,  to  express  its  power. 

Thus,  aa=za?,  aaa^=a^,  etc.  a™  indicates  that  a  is  taken 
as  a  factor  as  many  times  as  there  are  units  in  m. 

When  no  exponent  is  expressed,  1  is  understood.  Thus, 
a  is  the  same  as  a^,  each  signifying  the  Jirst  power  of  a. 

20.  A  Root  of  a  quantity  is  a  factor,  which,  multiplied 
by  itself  a  certain  number  of  times,  will  produce  the  given 
quantity. 

The  root  is  called  the  square  or  second  root,  the  cube  or 
third  root,  the  fourth  root,  etc.,  according  to  the  number 
of  times  it  must  be  taken  as  a  factor  to  produce  the  given 
quantity. 

Thus,  2  is  the  second  or  square  root  of  4,  since  2x2^4 ; 
a  is  the  fourth  root  of  a*,  since  aX«X«X«==^*- 

21.  The  Radical  Sign,  y^  or  |/,  when  prefixed  to  a 
quantity,  denotes  that  its  root  is  to  be  extracted. 

An  Index  is  a  figure  or  letter  placed  over  a  radical  sign 
to  denote  what  root  is  to  be  taken.     Thus, 

]^9,  or  ^9,  denotes  the  square  root  of  9,  which  is  3. 
1^8,  or  ^8,  denotes  the  cube  root  of  8,  which  is  2. 
I^a,  or  i^a,  denotes  the  fourth  root  of  a 

When  the  radical  sign  has  no  index  over  it,  2  is  under- 
stood ;  thus,  ^a  and  i^a  signify  the  same  thing. 

22.  An  Algebraic  Quantity,  or  an  Algebraic  Exprcs- 


DEFINITIONS.  1 5 

sion,  is  any  quantity  written  in  algebraic  language,  that  is, 
by  means  of  symbols.     Thus, 

5a,  is  the  algebraic  expression  of  5  times  the  number  a; 

36-)-4c,  is  the  algebraic  expression  for  3  times  the  number  b  in- 
creased by  4  times  the  number  c; 

3a2 — 7ab,  for  3  times  the  square  of  a,  diminished  by  7  times  the 
product  of  the  number  a  by  the  number  6. 

S3*  A  Monomial  is  a  quantity  nat  united  to  any  other 
by  the  sign  of  addition  or  subtraction ;  as,  4a,  a^hcj 
— 4icy,  etc. 

A  monomial  is  often  called  a  simple  quantity^  or  term. 

24.  A  Polynomial,  or  Compound  Quantity,  is  an  alge- 
braic expression  composed  of  two  or  more  terms ;  as,  a-\-h^ 
c — ic-f  y,  etc. 

25.  A  Binomial  is  a  quantity  having  two  terms  ;  as, 
a-f-^)  ^'^-fy?  etc. 

A  Residual  ftuantity  is  a  binomial,  the  second  term  of 
which  is  negative  ;  as,  a — h, 

26.  A  Trinomial  is  a  quantity  consisting  of  three 
terms;  as,  a-{-h — c. 

27.  The  Numerical  Value  of  an  algebraic  expression 
is  the  number  obtained  by  giving  a  particular  value  to  each 
letter,  and  then  performing  the  operations  indicated. 

Thus,  in  the  algebraic  expression  4a  —  Sc,  if  a=5  and 
c=^6,  the  numerical  value  is  4x5—3x6=20—18=2. 

28.  The  value  of  a  polynomial  is  not  affected  by  chang- 
ing the  order  of  the  terms,  provided  each  term  retains  its 
sign. 

Thus,  h"^ — 2ab-{-c  is  evidently  the  same  as  Z>^-f-c — 2ah. 

20.  The  Degree  of  any  term  is  equal  to  the  number 
of  literal  factors  which  it  contains. 


16  RAY'S  ALGEBRA,  SECOND  BOOK. 

Thus,  5a  is  of  the  first  degree;  it  contains  one  literal  factor. 

ax  is  of  the  second  degree;  it  contains  two  literal  factors. 
Sa^b^c='Saaabbc,  is  of  the  sixth  degree. 

30.  A  polynomial  is  said  to  be  homogeneous  when  each 
of  its  terms  is  of  the  same  degree.     Thus, 

a-{-b — 3c  is  homogeneous;  each  term  being  of  the  first  degree. 
x^ — 7x1/-  is  homogeneous ;  each  term  being  of  the  third  degree. 
X- — Sx^^  is  not  homogeneous. 

31.  An  algebraic  quantity  is  said  to  be  arranged  ac- 
cording to  the  dimensions  of  any  letter  it  contains,  when 
the  exponents  of  that  letter  occur  in  the  order  of  their 
magnitudes,  either  increasing  or  decreasing. 

Thus,  ax~-\-a'^X-~a^x^%  is  arranged  according  to  the  ascending 
powers  of  a;  and  bx"^ — b''''x'^-{-b'^x,  is  arranged  according  to  the  de- 
scending powers  of  X. 

32.  A  Parenthesis,  (  ),  is  used  to  show  that  all  the 
terms  of  a  polynomial  which  it  incloses  are  to  be  consid- 
ered together,  as  a  single  term.     Thus, 

10 — [a — b)  means  that  a — b  is  to  be  subtracted  from  10. 
h{a-\-b — c)  means  that  a-\-b — e  is  to  be  multiplied  by  5. 
5<x-|-(6— c)   means  that  b—c  is  to  be  added  to  5a. 

When  the  parenthesis  is  used,  the  sign  of  multiplication 

is  generally  omitted.  Thus,  (a — 6)X(«4-^)5  is  written 
(a—h){a^h). 

A  Vinculum,    ,   is  sometimes  used   instead    of  a 

parenthesis.  Thus,  a-{-hyC^h  means  the  same  as  b{a-\-h'). 
Sometimes  the  vinculum  is  placed  vertically;  it  is  then 
called  a  bar. 

Thus,     ax^,  is  the  same  as  (a — h-\-c')x'^. 
—h 

33.  Similar,  or  Like  Quantities,  are  such  as  contain 
the  same  letter  or  letters  with  the  same  exponents. 


DEFINITIONS.  17 

Thus,  3a6  and  —2ah,  Za^h  and  ba^h,  Za^h  and  — 5a^6, 
are  similar. 

Unlike  Quantities  are  such  as  contain  different  letters 
or  different  powers  of  the  same  letter. 

Thus,  ba  and  36,  Zab^  and  Sa^fe,  are  unlike  or  dissim- 
ilar. 

Remark. — An  exception  must  be  made  in  those  cases  where  let- 
ters are  taken  to  represent  coefficients.  Thus,  ax^  «nd  bx^  are  like 
quantities,  when  a  and  6  are  taken  as  coefficients  of  x^- 

34.  The  Reciprocal  of  a  quantity  is  unity  divided  by 
that  quantity.     Thus, 

The  reciprocal  of  a  is  — ;  of  3,  is  ^^ ;  of  |,  is  l-v-'^=4  ; 
Hence, 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 

35«  The  same  letter,  accented,  is  often  used  to  denote 
quantities  which  occupy  similar  positions  in  different  equa- 
tions or  investigations. 

Thus,  a,  a',  a",  a"\  read,  a,  a  prime,  a  second,  a  third, 
and  so  on. 

36.  The  following  signs  are  also  used,  for  the  sake  of 
brevity : 

00,  a  quantity  indefinitely  great,  or  infinity. 

.  * . ,  signifies  therefore^  or  consequently. 

• .  • ,  signifies  since,  or  because. 

r—'^  is  used  to  represent  the  difference  between  two  quan- 
tities, as  c, — 'cZ,  when  it  is  not  known  which  is  the  greater. 


EXERCISES. 

First,  copy  each  example  on  the  slate   or  blackboard ;  * 
and  then  read  it,  that  is,  express  it  in  common  language. 

Second,    find    the    numericnl   value    in   each,    supposing 
a=2,  6=^3,  c=4,  x=-h,  ^^6. 
2d  Bk.  2 


18 


RAY'S  ALGEBRA,  SECOND  BOOK. 
^     ex — a7/ 


1.  1b-^x—7/.  Ans.  20. 

2.  a'b^—Sx\  Ans.  —3. 

3.  c-^ay^c — a.  Ans.  10. 

4.  (c-|-a)(c— a.)  Ans.  12. 

5.  e!±^+£Z^.  Ans.  21. 


Ans.  4. 
Ans.  5. 


7    ^  ,  ^ 

o.  — ^ ^ — 1/ aby.  Ans.  v. 


9.  Find  the  difference  between  ahx^  and  a-\-b-\-x^  wlien 
a=4,  ?;=l,  rr— 3 ;  and  when  a=r5,  ^=7,  a;=12. 

Ans.  11  and  396. 

10.  Required  the  values  of  a2_j_2«5-f  ?*^  and  o^ — 2a5-|-i-, 
when  a=r:7  and  6=4.  Ans.  121  and  9. 

11.  What  is  the  value  of  n{n—V)  (n— 2)  (?i— 3),  when 
n=4,  and  when  w=10  ?  Ans.  24  and  5040. 

12.  Find  the  difference  between  ^ahc — 2a5,and  6o5c-=-2o5, 
when  a,  Z>,  c,  are  3,  5,  and  6  respectively.  Ans.  492. 


13.  Find  the  value  of 


7/^ 


a^^W 


when  a=5  and  6=3. 
Ans.  -,^. 


Verify  the  following,  by  giving  to  each  letter  any  value 
whatever  : 

14.  a{mA^ii){m — n)=om^ — an^. 


TO  BE  EXPRESSED  IN  ALGEBRAIC  SYMBOLS. 

1.  Five  times  a,  plus  the  second  power  of  b. 

2.  x^  plus  y  divided  by  Sz. 

3.  X  plus  y,  divided  by  3;^. 

4.  3  into  X  minus  n  times  y,  divided  by  m  minus  n. 

5.  a  third  power  minus  x  third  power,  divided  by  a  sec- 
ond power  minus  x  second  power. 

6.  The  square  root  of  m  minus  the  square  root  of  n. 

7.  The  square  root  of  m  minus  n. 


1.  5a+Z>2 


1.  x+. 


Zz 


ADDITION. 

ANSWERS. 

3.  "+^. 

Zz 

5. 

4     3x— Tiy 

6. 

7. 

19 


ADDITION. 

ST.  Addition,  in  Algebra,  is  the  process  of  finding  the 
simplest  expression  for  the  sum  of  two  or  more  algebraic 
quantities. 

There  are  three  cases  of  algebraic  addition  : 
1st.  "When  the  quantities  are  similar,  and  have  like  signs. 
2d.   When  the  quantities  are  similar,  but  the  signs  unlike. 
8d.   When  the  quantities  are  dissimilar,  or  part  similar 
and  part  dissimilar. 


38.  First  Case. — Let  it  be  required  to  find  the  sura 
of  Sx^y,  bx^y^  and  ^x'^y. 


Here,  x'^y  is  taken,  in  the  first  term,  3  times;  in 
the  second,  5  times;  and  in  the  third,  7  times; 
hence,  in  all,  it  is  taken  15  times.  Since  adding 
the  quantities  can  not  change  their  character,  and 
since  each  term  is  positive,  their  sum  is  positive. 

Find  the  sura  of  — Scc^y,  — 5x^y,  and  — ^x'^y. 

Here,  X^y  is  taken,  in  the  first  term,  — 3  times ;  in 
the  second,  — 5  times ;  and  in  the  third,  — 7  times ; 
hence,  in  all,  it  is  taken  — 15  times.  Therefore,  to 
add  similar  quantities  having  the  same  sign. 


OPERATION. 

-f  Ix'^y 
-f-]  ^x^ 


OPERATIOX. 

—  ^x'^^y 

—  Ix'^y 


— 15a:-?/ 

Rule. — Add  the  coefficients,  and  prefix  the  sum,  with  the 
common  sign,  to  the  literal  pari. 


20  RAY'S  ALGEBRA,  SECOND  BOOK. 

39.  Second  Case. — Let  it  be  required  to  find  the  sum 
of  -(-9a,  — 5«,   -i-4a,  and  — 2a. 

Here,  -\-9a-{~4a  is  -f  13a;  and  —5a— 2a  is  —7a.  operation. 

Now,    since   the   sum    of    two    equal    quantities,    of  -\-9a 

'which  one  is   positive    and   the    other   negative,   is  — 5a 

evidently  0,   — 7a  will  cancel   -l-7a  in  the  quan-  -|-4a 

tity  -(-13a,  and  leave   -(-6a  for  the  aggregate,  or  re-  — 2a 

suit  of  the  four  quantities.  ~ 

In  like  manner,  to  obtain  the  sum  of  — 9a,  -(-Sa, 
— 4a,  and  -(-2a,  we  find  the  sum  of  — 9a  and  —4a 

is  — 13a,  and  the  sum  of  -f  5a  and  -(-2a  is  -f-7a.  operation. 

Now,   -(-7a  will  cancel  — 7a  in  the  quantity  — 13a;  — 9a 

which  leaves  — 6a  for  the  aggregate.     Therefore,  -f  5a 

— 4a 
4- 2a 


—6a 
TO  ADD    SIMILAR   QUANTITIES    HAVING   DIFFERENT   SIGNS, 

Rule. — 1.   Add  the  positive  and  negative  coefficients  sep- 
arately. 

2.  Subtract  the  less  sum  from  the  greater ^  and  give  to  the 
difference  the  sign  of  the  greater. 

3.  Prefix  this  difference  to  the  literal  pai-t. 

40.  Third  Case. — Let  it  be  required  to  find  the  sum 
of  5a2— 8^>+c,   -\-h—a\  and   56+3a^ 

In  writing  the  quantities,  we  place,  for  con-  operation. 

venience,    those   which    are   similar   under   each  Sa^ — 8b-\-C 

other.  —  a--\-  b 

The  sum  in  the  first  column  is  -|-7a-,  and  in  3a2-|-56 
the  second,  — 26;  there  being  no  term  similar 
to  C,  it  is  annexed,  with  its  proper  sign. 


7a^— 26-l-c 
41«  From  the  preceding,  we  derive  the  following 


GENERAL    RULE   FOR  ADDITION   OF   ALGEBRAIC  QUANTITIES. 

1.    Write  the  quantities  to  he  added^  placing  those  that  are 
umiiar  under  each  other^ 


SUBTRACTION.  21 

2.  Add  the  similar  quantities  by  the  rules  already  given. 

3.  Annex  the  other  quantities  with  their  proper  signs. 

Remark. — In  algebraic  Addition,  Subtraction,  and  Multiplica- 
tion, it  is  best  to  begin  the  operation  at  the  left  hand. 

1.  Find   the   sum  of  4aa:-}-36^,    hax-\-^hy^    ^ax-\-Qhy, 
and  20ax-\-hy.  Ans.  37a£c-j-18%. 

2.  Of    10cz—2ax\    lhcz—Zax\     24cz—9ax\    and    Scz 


lax^.  Ans.  h2cz — 22ax 


3.  Of  3rcy— 10/,  — a;y-f  5/,  8xy— 6/,  and  4xy 
-1-2/.  Ans.  14a:y— 9/. 

4.  Of  a-fi-^c-ft?,  a-^t-{-c— J,  «-[-&— c-fcZ,  a—h^c 
-f  (Z,  and  _a+6-f-c-|-(Z.  Ans.  3a-|-36  +  3c+3(^. 

5.  Of  3(a;2— /),  8(a;2— /),  and  —h(x'~y''). 

Ans.  6(a;^ — y"^). 

6.  Of  10«2&  — 12a36c  — 156V-I-10,  —  4a26  +  8a36c 
_1062c*— 4,  — 3a26_3a36c-f 20Z>V— 3,  and  2a'^6-fl2a36c 
-f  55V-J-2.  Ans.  ba'h-\-Mhc-{-h. 

7.  Of  a"— Z;"-|-3x^,    2a'^—ZV—xP,    and    rt"*-!- 46'*— a;?. 

Ans.  4a"'-f  2ic^— a;«. 


SUBTRACTION. 


42.  Subtraction,  in  Algebra,  is  the  process  of  finding 
the  difference  between  two  algebraic  quantities. 

The  quantity  to  be  subtracted  is  called  the  subtrahend; 
that  from  which  the  subtraction  is  to  be  made,  the  minuend; 
the  quantity  left,  the  difference  or  remainder. 

The  explanation  of  the  principles  on  which  the  opera- 
tions depend,  may  be  divided  into  two  cases. 

1st.  Where  all  the  terms  are  positive. 

2d.  Where  the  terms  are  either  partly  or  wholly  negative. 


22  RAY'S  ALGEBRA,  SECOND  BOOK. 

43.  First   Case. — Let  it   be  required  to   subtract   4a 
from  7a. 

It  is  evident  that  7  times   any  quantity,  operation. 

less  4  times  that  quantity,  is  equal  to  3  times  la  Minuend, 

the  quantity;   therefore,  la  less  4a  is  equal  4a  Subtrahend, 

to  3a.  3a  Remainder. 

If  it  be  required  to  subtract  b  from  a,  we  operation. 

can  only  indicate  the  operation,  by  placing  a  Minuend, 

the  sign  minus  before  the  quantity  to  be  sub-  b  Subtrahend, 

tracted.  c^_6  Remainder. 


44.  Second  Case. — Let  it  be  required  to  subtract  h — c 
from  a. 

If  we  subtract  b  from  a,  the  result,  a — 6,  operation. 

is    obviously   too   little,    for   the   quantity   b  a       Minuend, 

ought  to  be  diminished  by  C  before  it  is  taken  b — e  Subtrahend. 

from  a.    We  have,  in  fact,  subtracted  a  quan-  q^ b-\-c  Remainder. 

tity  too  great  by  c,  and  to  obtain  a  true  re- 
sult, the  difference,  a — b,  must  be  increased  by  C;  this  gives,  for  the 
true  remainder,  a — b-\-C. 

To  illustrate  the  above  example  by  figures,  let  a=9,  6=5,  and 
C=S ;  and  let  it  be  required  to  subtract  5 — 3  from  9. 

The  operation  and  illustration  may  be  compared,  thus: 


From  a 
Take  b—c 

From  9                 .     . 
Take  5— 3            .     . 

.       .       r=9 

=2 

Rem.  a—b^c 

Rem.  9-5-1-3     .     . 

.     .     =7 

In  the  examples  already  explained,  the  same  result  would  have 
been  obtained  by  changing  the  signs  of  the  quantity  to  be  sub- 
tracted, and  then  adding  it. 

45.  From  the  preceding,  we  derive  the  following 


RULE  FOR  SUBTRACTION  OF   ALGEBRAIC   QUANTITIES. 

1.     Write    the    quantities^    placing    similar    terms    under 
each  other. 


SUBTRACTION.  23 

2.  Conceive  the  signs  of  all  the  terms  of  the  subtrahend  to 
he  changed,  from,  -j-  to  — ,  or  from  —  to  -\-^  and  then  pro- 
ceed hy  the  rule  for  algebraic  addition. 

Remark. — 1.  Beginners  should  solve  a  few  examples  hy  actually 
changing  the  signs  of  the  subtrahend. 

2.  Proof. — Add  the  remainder  and  the  subtrahend,  as  in  arithmetic. 

(1)  (1) 

From  Sa^b-Scx-  .H  The  ^ame,  with  f  Sa^b-Bcx-  z^ 

Take   3a2Z>4-4ca;— Sz^  V  the  signs  of  the  ^         —^a''b—4tcx-\-Zz' 

■ I   subtrahend   j  — — 

Kern,  ba'b—l cx-^-^z" )  changed.  VRem.  ba'b~lcx-[-2z'' 

(2)  (3) 

From       5a' — ^mz-\-hy*'  From  ax^ — 3c'/ — z^ 

Take  — 2a^-|-3mg-i-6/  Take  fca;^— 3r/-fy» 

Rem,       la^—6mz —  y  '  Kern,  (a— 6)x"^ —     f—z^ 

4.  From  4a— 26+3c  take  3a+4&— c. 

Ans.  a — 65+4c. 

5.  From  9a-,^— 4y+9  take  Trc^-f  5?/— 14. 

Ads.  2x^— 9y+23. 

6.  From  23a;/— 7y+lla;'  take  11a;/— 5^— Qx^. 

Ans.  12ic/— 2y-f  20x^ 

7.  From  12x+18  take  12a:— 18+^^.       Ans.  36—^. 

8.  From  x'—f  take  —4:—f-^4x\         Ans.  4— 3x^ 

9.  From  4ax^-\-hx-\-c  take  Sx^~2x-{-b. 

Ans.  (4a— 3)a;3-f  (2H-6)a:+c— 5. 

10.  From  — IT^c^+Qax^— ^a'ic+lSa'  take  — 19x'+9ax^ 
— 9a2a;+l7a^  Ans.  2x^-\-2a'x—2a\ 

11.  From  a^-\-Sx'-JrSx-\-l  take  x'^—Sx'^Sx—l. 

Ans.  6a;2-f  2. 

12.  From     9a^x'—lS+20ab'x—Wcx'     take      Sb^'cx'' 
-\-9a"'x'—6-\-3ab'x.        Ans.  11ab^x—1b"'cx'—^. 

13.  From    4a'^+2x^— »«    take    a*"— i"-l-3xP   and    2a"» 
^2b"—xP.  Ans.  a"^-f  46"— x«. 


24  RAY  S  ALGEBRA,  SECOND  BOOK. 

The  Bracket,  or  Vinculum. — As  the  Bracket,  or  Vin- 
culum, is  frequently  employed  in  relation  to  Addition  and 
Subtraction,  it  is  important  that  the  rules,  which  govern 
its  use,  should  be  well  understood. 

46.  1st.  Where  the  sign  phis  precedes  a  parenthesis,  or 
vinculum,  it  may  be  omitted  without  affecting  the  expression. 

This  is  self-evident,  as  is  also  the  converse,  viz. : 

Any  number  of  terms  may  be  inclosed  within  a  parenthesis 
preceded  by  the  sign  plus,  without  affecting  the  value  of  the 
expression. 

Thus.    a+(6— c)=a+6— c;    6+(5— 3)=6-|-5— 3,   and  a-f6— c 

-fd=a-f(6— c+d);   5+4—3+2=5+ (4—3+2). 

2d.  Where  the  sign  minus  precedes  a  vinculum,  it  may  be 
omitted  if  the  signs  of  all  the  terms  within  it  be  changed. 

For  the  minus  indicates  subtraction,  which  is  effected  by 
changing  the  signs  of  all  the  terms  of  the  quantity  to  be 
subtracted. 

Thus,  a—{b~c)—a — 6+c. 

a—{x—y^z)=a—x-^y-z. 

Sometimes  several  brackets,  or  vinculums,  are  employed 
in  the  same  expression,  all  of  which  may  be  removed. 

Thus,      a— {a+6— [a+6— c— (a— 6+c)]|, 
=a— {a+6— [a+6— c— a+6— c]}, 
=:a — \a-\-h — a — 6+c+a— 6+c}, 
=a — a — 6+a-j-6— c— a+6— c=6 — 2c. 

3d.  Any  quantity  may  be  inclosed  in  a  parenthesis  preceded 
by  the  sign  minus,  provided  the  signs  of  all  the  inclosed  terms 
be  changed. 

This  is  evident  from  the  preceding  principle. 

Thus,  a— 6+c=a— (6— c)=c— (6— a). 


OBSERVATIONS.  25 

This  principle  often  enables  us  to  express  the  same  quan' 
tity  under  several  different  forms. 

Thus,    a — 6-)-c-f  c/=ia— [6— c — d}, 

Simplify,  as  much  as  possible,  the  following  expressions; 

1.  (1— 2.T+3a:0  +  (3+2a:— a:^).  Ans.  4+2x1 

2.  ^a-h-c)-\-{h^c—d)-\-{d-e-^f)-^(e—f-^g). 

Ans    a — g. 

3.  3(:r^+/)- { {x'-\-2xy-^if)—(2xij—x^~f) } . 

Ans.  x^-\-y^^ 

4.  a — (x — «) — [x — (« — x)].  Ans.  3a — 3a;. 

5.  1— {1— [1— (1— a:)]}.  Ans.  X. 


OBSERVATIONS  ON  ADDITION  AND  SUBTRACTION. 

47.  All  quantities  are  to  be  regarded  as  positive,  unless,  for 
Bome  special  reason,  they  are  otherwise  designated.  Negative  quan- 
tities are  always,  in  some  particular  respect,  the  opposite  of  positive 
quantities.     Thus : 

If  a  merchant's  ffains  are  positive,  his  losses  are  negative;  if  lati- 
tude north  of  the  equator  is  -|-j  that  south  is  — •,  if  distance  to  the 
right  of  a  certain  line  is  -f ,  that  to  the  left  is  — ;  if  time  after  a 
certain  hour  is  -f?  time  be/ore  that  hour  is  — ;  if  motion  in  one 
direction  be  -f »  naotion  in  an  opposite  direction  is  — ;  and  so  on. 

48.  This  relation  of  the  signs  gives  rise  to  some  important 
particulars. 

1st.  TVie  addition,  to  any  quantity,  of  a  negative  number, 
produces  a  LESS  result  than  adding  zero. 

Thus, 


10 

10 

10 

10 

10 

10 

10 

3 

2 

1 

0 

-1 

2 

-3 

13 

12 

11 

10 

9 

8 

7 

It  will  also  be  seen,  from  this  illustration,  that  adding  a  negative 
number  produces  the  same  result  as  subtracting  an  equal  positive 
number. 

2d  Bk.  3* 


26"  RAY'S  ALGEBRA,  SECOND  BOOK. 

2(1    The   subtraction   of  a  negative  quantiti/  produces   a 
greater  result  than  subtracting  zero. 


Thus, 

10 

]0 

10 

10 

10 

10 

10 

3 

2 

1 

0 

—1 

—2 

—3 

7 

~8 

"9 

10 

11 

12 

13 

Here,  subtracting  a  negative  number  produces  the  same  result  as 
adding  an  equal  positive  number. 

49*  When  two  negative  quantities  are  considered  algebraically, 
that  is  called  the  least  which  contains  the  greatest  number  of  units; 
thus,  — 3  is  said  to  be  less  than  — 2,  But,  that  which  contains  the 
greatest  number  of  units  is  said  to  be  nuvierically  the  greatest ; 
thus,  — 3  is  numerically  greater  than  — 2. 

SO*  The  sum  of  two  positive  quantities  is  always  greater  than 
either  of  them.     Thus,  -|-5-|-3=-f8. 

The  sum  of  two  negative  quantities,  algebraically  considered,  is 
less  than  either  of  them.     Thus,  — 5 — 3= — 8. 

The  sum  of  a  positive  and  negative  quantity  is  always  less  than 
the  positive  quantity.     Thus,  -f  5 — 3=-[-2. 

51.  The  difference  of  two  positive  quantities,  as  in  arithmetic, 
is  always  less  than  the  greater  quantity.  Thus,  2a  from  ba  leaves 
3a,  or  5a— (+2a)=+3a. 

The  difference  of  two  negative  quantities  is  always  greater,  alge- 
braically considered,  than  the  minuend.  Thus,  — 2a  from  — 5a 
leaves  —3a,  or  — 5a— (— 2a)=— 3a. 

The  difference  between  a  positive  and  a  negative  quantity,  found 
by  subtracting  the  latter  from  the  former,  is  always  greater  than 
either  of  them.     Thus,  2a— (— a)=:3a 

1.  The  latitude  of  A  is  10°  N.  (+)  ;  the  latitude  of 
B  is  5°  S.  ( — )  ;  what  their  difference  of  latitude? 

Ans.  15°. 

2.  At  T  A.  M.,  a  thermometer  stood  at  —9°  ;  at  2  P.  M., 
at  4-15°  V  what  was  the  change  of  temperature  ? 

Ans.  24°. 


MULTIPLICATION.  27 


MULTIPLICATION. 

52.  Multiplication,  in  Algebra,  is  the  process  of  tak- 
ing one  algebraic  quantity  as  many  times  as  there  are  units 
in  another. 

The  quantity  to  be  multiplied  is  called  the  multiplicand; 
the  quantity  by  which  we  multiply,  the  multiplier;  and  the 
result,  the  product. 

The  multiplicand  and  multiplier  are  called  factors. 

53.  In  explanation  of  the  subject  of  algebraic  multipli- 
cation, we  begin  with  the  following 

Preliminary  Principle.— T'/ie  product  of  two  factors  is 
the  same,  whichever  he  made  the  midtiplier. 

To  prove  this,  suppose  we  have  a  sash  containing  a  vertical  and 
b  horizontal  rows. 

Since  there  are  a  vertical  rows  and  b  panes  in  each  row,  the 
whole  number  of  panes  will  be  represented  by  b  taken  a  times; 
that  is,  by  ab,  or  by  a  taken  b  times ;  that  is,  by  ba.  Hence,  ab  is 
equal  to  ba. 

In  a  similar  manner,  it  may  be  shown  that 
The  product  of  three,  or  of  any  number  of  factors,  is  the 
same,  in  whatever  order  they  are  taken. 

Thus,  a'Xh'Xc=ahc,  cab,  hac,  or  cha,  and  2x3x4=4 
X2x 3=3x2x4=4x3x2;  the  product  in  each  case 
being  24.     Also,  acy^6=^6ac,  or  6ca;  and  so  on. 

It  also  follows  from  this  principle,  that 

When  either  of  the  factors  of  a  product  is  multiplied,  the 
product  itself  is  multiplied. 

Thus,  2x3,  multiplied  by  5,  may  be  written  5x2x3, 
or  5x3x2  ;  that  is,  10x3,  or  15x2,  either  of  which  is 
equal  to  30. 

Remark. — The  distinction  between  the  multiplication  of  num- 
bers and  of  /actors  should  be  carefully  noticed.  Thus,  32X2=^4, 
but  3X2  multiplied  by  2  equals  6x2,  or  3x4. 


28  RAY'S  ALGEBRA,  SECOND  BOOK. 

54.  In  multiplication  there  are  four  things  to  be  con- 
sidered in  relation  to  each  term,  viz.  :  the  coefficient;  the 
literal  part;  the  exponent;  and  the  sign. 

55.  Of  the  Coefficient  and  Literal  Part. — 1.  Let  it 

be  required  to  find  the  product  of  2ac  by  8^. 

To  indicate   the  multiplication,  we  may  write     operation. 

the  product  thus,  2acX36.     But,  by  Art.  53,  this       2ac 

is  the  same  as  2x3X«^c,  and  2x3=6;  therefore,       36 

the  product  is  6a6c.     Hence,  ~    ~ 

6a0c  product. 

Rule  of  the  Coefficients. — Multiply  together  the  coefficients 
of  the  factors  for  the  coefficient  of  the  product. 

Rule  for  the  Literal  Part. — Annex  to  the  coefficient  all 
the  letters  of  the  factors  in  alphabetical  order. 


2.  3acX56=  l^ahc. 

3.  2amy^cn=         2acmn. 


4.  5aX4o'ic=         20aax. 

5.  7c3/x3y2;=:       21cy7/z. 


56.  Of  the   Exponent. — To  determine  the  rule  of  the 
exponents, 

1.  Let  it  be  required  to  find  the  product  of  2a'^  by  3a'. 

Since  2a^=z2aa,  and  3a^=3aaa^  the  product  operation. 

will  be  2aaX3««a,  or  Gaaaaa,  which,  for  the  2a^=2aa 

sake  of  brevity,   is  written    6a'*.     Hence,   we  Sa^=Saaa 

have  the  following  6a^=6aaaaa 

Rule  of  the  Exponents. — Add  the  exponents  of  any  letter 

in  the  factors  for  its  exponent  in  the  product. 


2.  ahXci=  a""}) 

3.  x^yXxy=  ir»/ 

4.  rj?x^zy^ax^=.        a*j?z^ 


5.  a'"Xa"=  «'"■'"'*. 

7.  x"'+^XaJ"~^=        a;'«+\ 


5'7.  From  the  two  preceding  articles,  we  derive  the  fol- 


lowing 


MULTIPLICATION.  29 


GENERAL  RULE  FOR  MULTIPLYING  ONE  POSITIVE  MONOMIAL 
BY    ANOTHER. 

1.  Multiply  the  coefficients  for  the  coefficient  of  the  product. 

2.  Annex  all  the  letters  found  in  both  factors. 

3.  When  the  same  letter  occurs  in  both,  add  its  exponents. 

1.  Multiply  be  hj  z Ans.  bcz. 

2.  Multiply  Sax  hj  by Ans.  Sabxy. 

3.  Multiply  4am  by  Sbn Ans.  12abmn. 

4.  Multiply  Mx  by  ^ax^y Ans.  85aVy. 

5.  Multiply  3tt'"ic"  by  9a"a;"*.      .     .  Ans.  27a"'+"a;"'+'*. 

58. — 1.  Required  to  find  the  product  of  a-\-b  by  m. 

Here,  the  sum  of  the  units  in  a  and  b  is  to  be         operation. 
taken  m  times.     The  units  in  a  taken  in  times         a-f  6 
:=.m,a^  and  the  units  in  h  taken  m  times  =^mb\         m 
hence,  a-\-b  taken  m  times  r^ma-^mb.     Hence,  , 

when  the   signs   are  positive,  we  have   the   fol- 
lowing 

Rule  for  Multiplying  a  Polynomial  by  a  Monomial. — 

Multiply  each  term  of  the  multiplicand  by  the  multiplier. 

2.  Multiply  x-{-y  by  w Ans.  wx-f-^y. 

3.  ax^-^-cz  by  3ar Ans.  Sa'^cx^-\Sac^z. 

4.  2a2+362  by  ^ab Ans.  lOa^fe+lSaR 

5.  mx-\-ny-\-vz  by  m^n.         Ans.  mhix-\-in^f^^y-\-'^^^^'^z. 

59. — 1.  Required  to  find  the  product  of  a+Z>  by  m-\-n. 

Here,  a-(-6  is  to  be  taken  as  many  times  as  there  are  units 
in  W-fn,  which  is  evidently  as  many  times  as  there  are  units 
in  m,  plus  as  many  times  as  there  are  units  in  n. 

Thus,  a+6 

ma-\-mb=  multiplicand  taken  m  times. 

na-\-nbz=  multiplicand  taken  n  times. 
»ia-f  77i6-j-wa4-n6=  multiplicand  taken  {m-{-n)  times. 


30  RAY'S  ALGEBRA,  SECOND  BOOK. 

Hence,  when  the  signs  are  positive,  we  have  the  following 

Rule  for  Multiplying  one  Polynomial  by  Another. — 

Multiply  each  term  of  the  multiplicand  hy  each  term  of  the 
multiplier^  and  add  the  products. 

2.  Multiply  x-\-y  by  a+c.     .     .    Ans.  ax-\-ay-\-cx-^cy. 
8.       2x-\-Sz  by  Sx-{-2z.    .     .     .    Ans.  6x'-\-lSxz+6z\ 

4.  2aH-c  by  a+2c Ans.  2a'-^bac-\-2cK 

5.  x^-\-xy-{-y^  by  x-j-y.  .     .  Ans.  x^-^2x'^y-\-2xy^-\-y^. 

6.  a^+2ai+&'  by  a+b.     .  Ans.  a^+Sa'b+Sa¥+h\ 

60«  Of  the  Signs. — In  the  preceding  article  it  was  as- 
sumed that  the  product  of  two  positive  quantities  is  posi- 
tive. The  general  rule  for  this,  and  the  other  cases  which 
may  arise  in  algebraic  multiplication,  may  be  deduced,  as 
follows : 

1st.  Let  it  be  required  to  find  the  product  of  +^  by  a. 

The  quantity  6,  taken  once,  is  -{-b;  taken  twice,  is  -|-26;  taken 
3  times,  is  -{-Sb;  and  hence,  taken  a  times,  it  is  -\-ab. 

Hence,  the  product  of  two  positive  quantities  is  positive;  or,  more 
briefly  expressed,  plus  multiplied  by  plus  gives  plus. 

2d.  Let  it  be  required  to  find  the  product  of  — b  by  a. 

The  quantity  — 6,  taken  once,  is  — 6;  taken  twice,  is  — 26; 
taken  3  times,  is  — 36;  and  hence,  taken  a  times,  is  — ab. 

Hence,  a  negative  quantity,  multiplied  by  a  positive  quantity,  gives 
a  negative  product;  or,  more  briefly,  minus  multiplied  by  plus  gives 


3d.  Let  it  be  required  to  multiply  b  by  — a. 

Since  by^-^a  implies  that  b  is  to  be  added  a  times  to  0,  6X — ct 
must  indicate  (Art.  47)  that  6  is  to  be  subtracted  a  times  from  0.  Sub- 
tracted once,  it  is  — 6;  subtracted  twice,  — 26;  and  so  on.  Hence, 
subtracted  a  times,  it  is  — ab. 

Therefore,  a  positive  multiplied  by  a  negative  quantity,  gives  a 
negative  product;  or,  plus  multiplied  by  minus  gives  minus. 


MULTIPLICATION.  31 

4th.  Let  it  be  required  to  multiply  — h  by  — a. 

Reasoning  as  above,  — 6  subtracted  a  times  from  0,  gives  -\-ab. 
Hence,  the  product  of  two  negative  quantities  is  positive;  or,  more 
briefly,  minus  multiplied  by  minus  gives  plus. 

Note. — The  following  proof  of  the  3d  and  4th  cases  is  generally 
regarded  as  more  satisfactory  than  the  preceding. 

Let  it  be  required  to  find  the  product  of  c — d  by  a — h. 

Here  it  is  required  to  take  e — d  as  many  times  as  there  are  units 
in  a — b.  This  will  be  done  by  taking  C — d  as  many  times  as  there 
are  units  in  a,  and  then  subtracting,  from  this  product,  G — d  taken 
as  many  times  as  there  are  units  in  6. 

Thus,  c — d 
a—b 
ac — ad=c—d  taken  a  times, 

be — bdz=c — d  taken  b  times. 

ac—ad—bc-\-bd.  By  subtraction,  =c — d  taken  a — 6  times. 

The  final  result,  in  the  terms,  — 6c  and  -\-bd,  is  what  it 
would  have  been  if  we  had  added  the  partial  products,  assuming 
that  -\-C  multiplied  by  — b  gives  — 6c,  and  that  — d  multiplied 
by  — 6  gives  -\-bd.  As  we  know  the  result  to  be  correct,  we  infer 
that  the  assumption  would  be  correct,  viz.:  that  plus  by  minus  gives 
minus^  and  minus  by  minus  gives  plus. 

From  the  above,  we  derive  the  following 


General  Rule  for  the  Signs. —  The  product  of  like  signs 
gives  plus,  and  of  unlike  signs,  minus. 


GENERAL    RULE    FOR   THE    MULTIPLICATION   OF   ALGEBRAIC 
QUANTITIES. 

61. — 1.  Multiply  every  term  of  the  multiplicand  hy  each 
term  of  the  multiplier,  observing  the  rules  for  the  coefficients, 
the  exponents,  and  the  signs. 

2.  Add  the  several  partial  products  together. 


32  RAY'S  ALGEBRA,  SECOND  BOOK. 


NUMERICAL  EXAMPLES  TO  VERIFY  THE  RULE  OF  THE  SIGNS. 

1.  Multiply  V-4  by  5.    Ans.  35-20=.15=:3x5. 

2.  84-3  by  6-4.  Ans.  48-14— 12=22==llx2. 


GENERAL    EXAMPLES. 

1.  Multiply  4a2_3ac-f  2  by  ^ax. 

Ans.  20a'x-lbah'x-\-10ax. 

2.  ba—2ah-\~10  by  —9ah. 

Ans.  ~4ba'h-{-lSa'h'—90ah. 

3.  2x-{-Sz  by  2x—Sz.  Ans.  ^x'-9z\ 

4.  4a2_6a-f9  by  2a+3.  Ans.  Sa'+2l. 
5         a — h-^c — d  by  a-\-h — c — d. 

Ans.  a'—h^—c^^d'—2ad'{-2bc. 

6.  a^-\-f-\-z'  by  x'+f. 

Ans.  a:^4-a-.y+.TV4-a:y+/+/;22^ 

7.  a'-\-Sa'h-\-Sah''-\-h'  by  a3_3a2^^3a^,2_2,3^ 

Ans.  a^— 3a*i2+3a2i*_Z,6. 
8        12x'—Sx'i/-\-Uxf—10f  by  3a:H-2y. 

Ans.  36a:*+29a;y— 20/. 
9.       a^+a.'K+ic^  by  a/^ — ax-{-x'^.      Ans.  a*-|-a^a;^+a;*. 

10.  a'-i-2ah-{'2h^  by  a2_2a^;+262.     Ans.  a*+4?>*. 

11.  l-\-x-^x'^-{-a^'{-x*  by  1 — x.  Ans.  1 — a;^. 

12.  27a^-\-9x'7/-hSxf-]-f  by  3a;— 3/.  Ans.  81a^*— /. 

13.  a'-\-2a'b-{-2ab''^h'  by  a"— 2a2&  +  2a7>2_Z,3^ 

Ans.  a^ — 6*. 

14.  x'—a^-\-x^—x+l  by  ir'+a^— 1. 

Ans.  rr" — x*-{-a^ — x'^-}-2x — 1. 

15.  l-i-x-{-x*-\-x^  by  1 — x-\-x^ — ar*.        Ans.  1 — a^. 

16.  Multiply  together  x — 3,  x-\-4,  x — 5,  and  x-\-6. 

Ans.  a;*+2a;»— 41x2— 42a:+360. 

17.  a+t,  a—h,  a^-\-ah+h\  and  a'—ah+h-. 

Ans.  tt'' — 6^ 


MULTIPLICATION.  33 

62.  Multiplication  by  Detached  Coefficients. — In  tlie 
multiplicatiou  of  polyooiiiials,  it  is  evident  that  the  coeffi- 
cients of  the  product  depend  on  the  coefficients  of  the  fac- 
tors, and  not  upon  the  literal  parts  of  the  terms. 

Hence,  by  detaching  the  coefficients  of  the  factors  from, 
the  literal  parts,  and  multiplying  them  together,  we  shall 
obtain  the  coefficients  of  the  product.  If  to  these  coeffi- 
cients, the  proper  letters  are  then  annexed,  the  whole  prod- 
uct will  be  obtained.  This  method  is  applicable  where  the 
powers  of  the  same  letter  increase  or  decrease  regularly. 

1.  Multiply  a'— 2a6-(-Z>2  by  a-\-h.  orERATioN. 

1-2-1-1 

After  finding  the  coeflBcients,  it  is  obvious         ji  j 

that  a^  will  be  the  first  term,  and  b^  the  last         -, f,  ,  ^ 

term;    hence,  the  entire  product  is  a^ — a^b  ,  ,     ^  .  ^ 

— ab^-\-b^. 


1_1_1_^1 


2.  Multiply  a'—Sa'h-j-h^  by  a'—h\ 


In  this  example,  supposing  the  powers  operation 

of  a  to  decrease   regularly   toward  the  1 — 3-|-0-f  1 

left,   it  is  obvious  that  there  is  a  terra  1-j-O — 1 

wanting  in  each  factor.     These  must  be  -[ 3  i  oij 

supplied   by  0.     The  entire  product   is  l-f3 0 1 

a^—Sa^  b  -a^b^-]  Aa-b^—b->.  i_3_i  1  4_o_i 

3.  Multiply  m^ -\-mhi-\-'mn'^ -\-7i^  by  m — n.   Ans.  m* — n^. 

4.  Multiply  l-]-2z-\-Sz'^4z'-\-b^  by  1—z. 

Ans.  l-i^z-{-z^-\-^-^z'—^J. 

By  this  method,  let  the  general  examples.  Art.  61,  from  7  to  14  in- 
clusive, be  solved. 

REiMARKS   ON   ALGEBRAIC   MULTIPLICATION. 

OS«  The  degree  of  the  product  of  any  two  monomials  is  equal 
to  the  sum  of  the  degrees  of  the  multiplicand  and  multiplier. 
Thus,  2a-b,  which  is  of  the  3d  degree,  multiplied  by  oab^  of  the  4th 
degree,  gives  ^W'b^^  which  is  of  the  7th  degree. 


34  RAY'S  ALGEBRA,  SECOND  BOOK. 

This  is  also  true  of  two  polynomials;  as  an  illustration  of  which, 
see  Example  7,  Art.  61. 

04:«  In  the  multiplication  of  two  polynomials,  when  the  partial 
products  do  not  contain  similar  terms,  if  there  be  m  terms  in  the 
multiplicand,  and  n  terms  in  the  multiplier,  the  number  of  terms 
in  the  product  will  be  nvyn.  Thus,  in  Example  6,  Art.  61,  there 
are  3  terms  in  the  multiplicand,  2  in  the  multiplier,  and  3x2=6  ia 
the  product. 

6S«  If  the  partial  products  contain  similar  terms,  the  number  of 
terms  in  the  reduced  product  will  evidently  be  less  than  my^^n;  see 
Examples  7  to  18  inclusive.  Art.  61. 

06*  When  the  multiplication  of  two  polynomials,  indicated  by 
a  parenthesis,  as  (m-\-n)[p — g),  is  actually  performed,  the  expres- 
sion is  said  to  be  expanded,  or  developed. 


DIVISION 


OT.  Division,  in  Algebra,  is  the  process  of  finding 
how  many  times  one  algebraic  quantity  is  contained  in 
another. 

Or,  having  the  product  of  two  factors,  and  one  of  them 
given,  Division  teaches  the  method  of  finding  the  other. 

The  quantity  by  which  we  divide  is  called  the  divisor; 
the  quantity  to  be  divided,  the  dividend;  the  result  of  the 
operation,  the  quotient. 

68.  In  division,  as  in  multiplication,  there  are  four 
things  to  be  considered,  viz. :  the  sign;  the  coefficient;  the 
exponent;   and  the  literal  part. 

60.  To  ascertain  the  rule  of  the  siprns. 


Since,  +«X+6=+a6  ^ 

-aX+6=:-a6  I   therefore, 
-f  ax— 6=— a6  I 
— ax— 6=+a6  J 


)4-a6H-H-6^+a 
— a6-^+6=— a 
+  a6-= — br=—a 
—ab-i — 6=4-a 


DIVISION.  35 

From  the  foregoing  illustration,  we  derive  the  following 

Rule  of  the  Signs. — Like  signs  in  the  divisor  and  divi- 
dend give  plus  in  the  quotient;  unlike  signs  give  minus. 

TO.  The  rule  of  the  coefficients,  the  rule  of  the  exponents^ 
and  the  i-ide  of  the  literal  part,  may  all  be  derived  from  the 
solution  of  a  single  example. 

Required  to  find  how  often  2a^  is  contained  in  6a^h, 

2a^      2 

Since  division  is  the  reverse  of  multiplication,  the  quotient  mul- 
tiplied by  the  divisor,  must  produce  the  dividend;  hence,  to  obtain 
this  quotient,  it  is  obvious, 

1st.  That  the  coefficient  of  the  quotient  must  be  such  a  number, 
that  when  multiplied  by  2  the  product  shall  be  6  ;  therefore,  to  obtain 
it,  we  divide  6  by  2.     Hence,  the 

Rule  of  the  Coefficients. — Divide  the  coefficient  of  the 
dividend  by  the  coefficient  of  the  divisor. 

2d.  The  exponent  of  a  in  the  quotient  must  be  such  a  number, 
that  when  2,  the  exponent  of  a  in  the  divisor,  is  added  to  it,  the 
sum  shall  be  5;  that  is,  it  must  be  3,  or  5 — 2.     Hence,  the 

Rule  of  the  Exponents.  —  Suhtract  the  exponent  of  any 
letter  in  the  divisor  from  the  exponent  of  the  same  letter  in  the 
dividend  for  its  exponent  in  the  quotient. 

3d.  The  letter  b,  which  is  a  factor  of  the  dividend,  but  not  of  the 
divisor,  must  be  in  the  quotient.     Hence,  the 

Rule  of  the  Literal  Part. —  Write,  in  the  quotient,  every 
letter  found  in  the  dividend,  and  not  in  the  divisor. 

Tl.  The  preceding  rules,  taken  together,  give  the  fol- 
lowing 


36 


RAYS  ALGEBRA,  SECOND  BOOK. 


GENERAL  RULE  FOR  DIVIDING  ONE  MONOMIAL  BY  ANOTHER. 

1.  Prefix  the  proper  sign,  on  the  ptinciple  that  like  signs 
give  plus,  and  unlike  signs  give  minus. 

2.  Divide  the   coefficient  of  the   dividend    hy  that  of  the 
divisor. 

3.  Subtract  the  exponent  of  the  divisor  from  that  of  the 
dividend,  when  the  same  letter  or  letters  occur  in  both. 

4.  Annex  any  letter  found  in  the  dividend  but  not  in  the 
divisor. 


1.  Divide  4a^  by  2a''  and  by  —2a' 

2.  SOa*b'  by  ^a'b.   .     .     . 

3.  — 28xy2*  by  —Ixyh. 

4.  —Sba^b^c  by  bab\  .     . 

5.  S2xyz  by  — Sxy.      .     . 

6.  42c^m^7i  by  — Scmn. 

7.  £c'"+"  and  ic"»-"  each  by  x 

8.  1;'"+"  by  v'^-^p.       .     .     . 


Ans.  2a'  and  — 2a^ 
.  .  Ans.  6a^b' 
.  Ans.  4xYz\ 
.  Ans.  — 7abc. 
.  .  Ans.  — 4:z. 
Ans.  — 14c^m. 

Ans.  x""  and  x'"~'^\ 
.     .     .    Ans.  v""-^. 


Note. — In  the  following  examples,  the  quantities  included 
within  tLe  parenthesis  are  to  be  considered  together,  as  a  single 
quantity 

9.  Divide  (a-f  6)'  by  (a-\-by.  .     .     -.     .    Ans.  (a+6). 

10.  (m — ny  by  (m — ny Ans.  (m — w)'. 

11.  S(a—byx'y  by  2(a—b)xy.     ,     .   Ans.  4(a—byx. 

12.  (a-{-bx'y+^  by  (ia+bx')P-\   .     .    Ans.  (^a+bx'y. 

•72.  It  is  evident  that  one  monomial  can  not  be  divided 
by  another  in  the  following  cases : 

1st.  When  the  coefficient  of  the  dividend  is  not  exactly 
divisible  by  the  coefficient  of  the  divisor. 

2d.  When  the  same  literal  factor  has  a  greater  exponent 
in  the  divisor  than  in  the  dividend. 

3d.  When  the  divisor  contains  one  or  more  literal  fac 
tors  not  -^ound  in  the  dividend. 


DIVISION.  37 

In  each  of  these  cases  the  division  is  to  be  indicated  by 
a  fraction.     See  Art.  119. 

73.  It  has  been  shown,  Art.  53,  that  any  product  is 
multiplied  by  multiplying  either  of  its  factors ;  hence,  con- 
versely, any  dividend  will  he  divided  hy  dividing  either  of 
its  factors. 

Thus,  6x9^3r=2x9;  or,  6x3=18. 

T4.  Division  of  Polynomials  by  Monomials. — In  mul- 
tiplying a  polynomial  by  a  monomial,  we  multiply  each  term 
of  the  multiplicand  by  the  multiplier.  Hence,  conversely, 
we  have  the  following 


RULE    FOR    DIVIDING    A    POLYNOMIAL   BY    A    MONOMIAL. 

Divide  each  term  of  the  dividend  hy  the  divisor^  accord- 
ing to  the  rule  for  the  division  of  monomials. 

Note. — Place  the  divisor  on  the  left,  as  in  arithmetic. 

1.  Divide  a}-\-ah   by  a Ans.  a  -{-h. 

2.  '6xy-\-2x^y  by  — xy Ans.  — 3 — 2x. 

8.  10ah—lbz'—2bz  by  5^.  .  .  .  Ans.  2a'—Sz—6. 

4.  Sah-]-12ahx—9a''h   by  —Sah.    Ans.  —l—4x-^Sa. 

5.  5x^y — 40aV3/^-|-25a*icy  by  — ^xy. 

Ans.  — x^y'^-\-Sa'^xy — 5a*. 

6.  4:ahc—2iah^—S2ahd  by  — 4a6. 

Ans.  —c-{-6b-i-Sd. 

7.  a'"Z>3+a'"+iZ>2+a"-26  by  ah. 

Ans.  a'«-^6'+a'"6+a"-3. 

8.  Sa(^x-\-y)-]-c\x-]-yy   by  x-\-y.    Ans.  Sa+cXx-^y). 

9.  (h-\-c)(h—c^'—{h—c)(h+cy   by  (6-|-c)(&— c). 

Ans.  (fe— c)— (6-f  c)=— 2c. 
10.  h^c(m-{-n)—hc\m-{-n)   by  5c(m-f  ?i)   Ans.  b—c. 


38  RAY'S  ALGEBRA,  SECOl^D  BOOK. 


DIVISION  OF  ONE  POLYNOMIAL  BY  ANOTHER. 

•75.  To  deduce  a  rule  for  the  division  of  polynomials,  we 
shall  first  form  a  product,  and  then  reverse  the  operation. 


Multiplication,  or  formation 
of  a  product. 

a3_5cx26 

a^—ba^b 

-o-'b^^^a^b^ 

a^— 3a46_i  1  aW-\-ba^b'^ 


a^^2ab~b'^ 
Quotient. 


Division,  or  decomposition  of  a  product 

a5_3a46-lla362-j-5a263  a3_5a26 
i.stii+2a<6^  1  Ia362_^5a2^)3 

2d  Remainder,   — tt^ft^-f-Sa^ft^ 

— a362^5a2^>3 

'.'A  Remainder,  0 


The  dividend,  or  product,  and  the  divisor,  being  given,  (Art.  67),  it 
is  now  required  to  find  the  quotient,  or  the  other  factor. 

This  dividend  has  been  formed  by  multiplying  the  divisor  by  the 
several  terms  of  the  quotient,  and  adding  the  partial  products  to- 
gether. These  several  unknown  terms,  constituting  the  quotient, 
we  are  now  to  find. 

Arranging  the  dividend  and  divisor  according  to  the  decreasing 
powers  of  the  letter  a,  it  is  plain  that  the  division  of  a'',  the  first 
term  of  the  dividend,  by  a^,  the  first  term  of  the  divisor,  will 
give  a^,  the  first  term  of  the  quotient. 

If  we  subtract  from  the  dividend  a''— 5a^6,  which  is  the  product 
of  the  divisor  a^—5a^b  by  a^,  the  first  term  of  the  quotient,  the 
remainder  -\-2a^b—lla^b^-\-5a-b^,  will  be  the  product  of  the  divisor 
by  the  other  terms  of  the  quotient. 

The  first  term  -fSa^^  of  the  Ist  remainder,  is  the  product  of 
the  1st  term  a^  of  the  divisor  by  the  1st  of  the  remaining  unknown 
terms  of  the  quotient;  therefore,  we  shall  obtain  the  2d  term  of  the 
required  quotient,  by  dividing  -f2a'*6  by  a^;  this  gives  -f2a6. 

Multiplying  the  divisor  by  -\-2ab,  and  subtracting  the  product, 
we  have  a  2d  remainder,  which  is  the  product  of  the  divisor  by  the 
remaining  term  or  terms  of  the  quotient;  hence,  the  division  of 
the  1st  term  —a^b^  of  this  2d  remainder,  by  the  1st  term  a^  of  the 
divisor,  must  give  the  3d  term  of  the  quotient,  which  is  found  to 
be  —62. 

The  remainder  zero,  shows  that  the  quotient  a2-f2a6— 62  is  exact, 
since  the  subtraction  of  the  three  partial  products  has  exhausted  the 
dividend. 


DIVISION. 


39 


It  is  immaterial  whether  the  divisor  be  placed  on  the  right  or  left 
of  the  dividend ;  by  placing  it  on  the  right,  it  is  more  easily  multi- 
plied by  the  respective  terms  of  the  quotient. 

TO.  From  the  above,  we  derive  the  following 


RULE  FOR  THE  DIVISION  OF  ONE  POLYNOMIAL  BY  ANOTHER. 

1.  Arrange  the  dividend  and  divisor  with  reference  to  a 
certain  letter. 

2.  Divide  the  first  term  of  the  dividend  hy  the  first  term 
of  the  divisor^  for  the  first  term  of  the  quotient.  Multiply 
the  divisor  hy  this  term,  and  subtract  the  product  from  the 
dividend. 

3.  Divide  the  first  term  of  the  remainder  hy  the  first  term 
of  the  divisor,  for  the  second  term  of  the  quotient.  Multiply 
the  divisor  hy  this  term,  and  subtract  the  product  from  the 
last  remainder. 

4.  Proceed  in  the  same  mariner,  and  if  the  final  remainder 
is  0,  the  division  is  said  to  he  exact. 

1.  Divide  15x'+16x^— 15/  by  ^x—Sy. 

OPERATION. 

15x^-]-16xy-lby^\5x-Sy 

Ibx^—  9xy  ^x\-by,  Quotient. 

4-25a:.v— 15?/2 


2.  Divide  m'^ — n'^  by  m-^-n. 

OPERATION. 

rri'—n^      |m-fn 

^2_j_77in      m— ^  Quotient. 

— mn — 7i2 
— Tnn— ^2 


3.  Divide  a;'-fy  by  x-\-y. 

OPERATION. 

x^-^y^    \x^y 

X'^ArX^y        x^—xy\y',   Quot. 

— a;22/+  2/3 
—x^y—xy"^ 

J^xy^^y^ 
xy'^^y^ 


40 


RAYS  ALGEBRA,  SECOND  BOOK. 


4.  Divide  1x'y-\-bxy''-\-2x^-^y^  by  Zxy-\-x'^y\ 

Arranging  the  divisor  and  dividend  with  reference  to  a;,  we  have 
the  following: 

OPERATION. 

2x^^lx^y^bxy'^^y^     \x'^j^^xy^y'^ 
2x'^-\-Qx'^y^2xy'^  '•2x-\y,  Quotient. 


5.  Divide  x^-\-x^ — ^x^-\hx^  by  x — x^. 


Division  performed,  by  arranging  both 
quantities  according  to  the  ascending 
powers  of  x. 


Division  performed,  by  arranging  both 
quantities  according  to  the  descendini/ 
powers  of  x. 


x^-\-x^—7x^-\-dx-'\x--x^ 

Bx'^—7x*^x^-\-x2\  —x^^x 

x'^—x^                   x-y2x^—dx^, 

hx'^-hx^              —hx^y2x^-^x, 

2a;3— 7a:*                     Quotient. 

— 2xt-f  x?-                 Quotient. 

2a:3_2a:4 

-2a:4+2a:3 

— 5a:4+5a:5 

— a:3-f  a;2 

—5x*-^dx-> 

-x^^x^ 

The  two  quotients  above  are  the  same,  but  differently  arranged. 


6.  Divide  ^x^-\-bxy — 4j/^  by  Zx-\-^y.      Ans.  2x — y. 

7.  x^— 40a:— 63  by  a:— 7.      Ans.  a:'^-|-7a;-|-9. 

8.  3/i^H-16/i*A:-33/tVi:^-f-14/iV^3  ^y  A2_|_7/j,. 

Ans.  Zh^—bK'h-^2hlc\ 

9.  a^— 243  by  a— 3.  A.  «*+3a»-f-9a2-f  27a-|-81. 

10.  a:«— 2a'a;''-j-««  by  x'—2ax-\-a} 

Ans.  a;*+2aa:^-f  3«V4-2a'a:+a*. 

11.  1— 6a:5-f-5x«  by  l_2a:+a:^ 

Ans.  l-j-2.'r+3a:H-4a;3-h5^*- 

12.  P^-\-p^l-\-'^P^' — 2q'^-\-7qr — 3/'^  by  p — ^-f  3r. 

Ans.  p-\-2q — r. 

13.  4a:54-4a:— a:^  by  3.^+23:^+2. 

Ans.  2x'—Sx'-^2x. 

14.  ic*— a«  by  a^-^2ax''-j-2a'x-{-a\ 

Ans.  a^—2ax''-j-2a'x—a\ 


DIVISION.  41 

15.  Divide  m^-(-2mp — n"^ — 2/«g-f-i^^ — 2^  ^y  '^^ — n-\-p — q. 

Ans.  m-\-n-\-j)-\-q. 

16.  a^^b'-\~c^—Sabc  by  a+6+c. 

Ans.  a"^.-[-6"^-|-c2 — ab — ac — be. 

17.  a;"'+"-f-.x"yM-*'^"'3/"''-fy"''""  by  a;"-f^"*.  A.  a;"'-]-^". 

18.  aa;^ — (a'^-\-b)x^-\-b^  by  ao; — Z>.    Ans.  x'^ — ax — b. 

19.  a2m_3^»«^«_|.2c2«  by  a"*— c".         Ans.  a"* — 2c». 

20.  x*-[-^  * — a:^ — a:"^  by  x — x'^.  Ans.  x^ — x'^. 

21.  a«4-a«6--f  a*6*-fa2^>6_|_58  ^y  a'^a'b^a'b'-\^ah'-}~b\ 

Ans.  a*— a^t-fa'-^/;-^— aZ/3-fZ,*. 

22.  a^-f  (a— l)x-+(a— l>3  +  (rt  — l)a;*  — .T.s     by 

a — a;.  Ans.  a-\-x-{-x'^-\-x^-\-x*. 

23.  1— 9a;8— 8.^»  by  1+2.^+.^'^ 

Ans.  1  —  2x-\-  Sx'—4x^-{-  ^x*—6x'-{-  1x^—Sx\ 

24.  l-|-2x  by  1 — Sx  to  5  terms  in  the  quotient. 

Ans.  l+5a;+15a;^+45a;3-|-135a;*-|-  etc. 

77.  Division  by  Detached  Coefficients. — Division  may 
sometimes  be  conveniently  performed  by  detaching  the 
coefficients,  as  explained  in  Art.  62.     Thus, 

1.  Let  it  be  required  to  divide  x'^-{-2xi/-^i/^  by  x-\-i/. 

l-fS-fljl+l  Hence,   the    coefficients    of    the   quotient 

1-fl        1+1  are  1  and  1.    A\so,x^-ir'X=x,a.ndy^-^y—y; 

-|-l-fl  therefore,  the  quotient  is  Ix^ly,  or  x-\^y. 

2.  Divide  12a'—2Qa'b—Sa'b'-{-10ab'—Sb'  by  Sa^— 2a& 

12—26—8+10—813—2+1  Hence,  the  coefficients  of  the 

12—  8+4               4—6—8  quotient  are  4—6—8.     Also, 

—18-12+10  a^--a2=a2^  and  b*^b^=b^) 

—18+12—  6  therefore,  the  quotient  is  ia^ 

—24+16—8  _6a6— 862. 

—26+16-8 


2d  Bk. 


42  RAY'S  ALGEBRA,  SECOND  BOOK. 


1.  Divide 

a^+a:^  by  a+a 

;. 

1+0+0+1 
1  +  1 

11+1 
1-1+1 

—1 

+1+1 
+1+1 

a2_aa:+x2, 

Quotient. 

4.  Divide    w^ — 6m*n-\-10mbi^ — 10mV-j-5m7i* — 7i*    by 
m' — 2mn-\-n^.  Ans.  77i^ — 8??i^n+3m»^ — «^. 

5.  ©ivide  a^—Za'h'-\-2>a'h'—h^  by  a^— 3a^Z>+3a6'^— //. 

Ans.  a?-\-^d'h^^al'-^h\ 


Most  of  the  examples  in  Art.  76  may  be  solved  by  this  method. 


II.    ALGEBRAIC    THEOREMS, 

DERIVED    FROM    MULTIPLICATION    AND    DIVISION. 

Remark — One  of  the  chief  objects  of  Algebra  is  to  establish 
certain  general  truths.  The  following  theorems  serve  to  show  some 
of  its  most  simple  applications. 

TS.  Theorem  I. —  The  square  of  the  sum  of  two  quantities 
is  equal  to  the  square  of  the  first,  plus  twice  the  product  of  the 
first  hy  the  second,  plus  the  square  of  the  second. 

Let  a  represent  one  of  the  quantities  and  6         a  +5 
the  other.  a  +6 

Then,a+6=  their  sum;  and  (a+6)X(«+6),  «^+  «& 

or  (a+6)2=  the  square  of  their  sum.    By  mul-  +  a6+62 

tipjying,  we  obtain  a^-\-2ab-\-b-,  which  proves  a~^2ab^i^ 
the  theorem. 

APPLICATION. 

i:  (2+5)2=4+20+25:=49. 

2.  (2w+3»)2=:4//i2+i2wn+9?i2. 


ALGEBRAIC  THEOREMS.  43 

4.  (aa;2-}-3a;2;3)2==a2a;*-t- 6aa;32;3_|_9a;22;6. 

TO.  Theorem  II. — Hie  square  of  the  difference  of  two 
quantities  is  equal  to  the  square  of  the  first,  minus  twice  the 
product  of  the  first  hy  the  second,  plus  the  square  of  the 
second. 

Let  a  represent  one  of  the  quantities,  and  b         a  — b 
the  other.  a  — b 


Then,  a — 6=  their  difference;  and  (a — 6)X  <^^ —  «^ 

{a — 6),  or  [a—bY=  the  square  of  their  dif-  —  ab-^b"^ 

ference.     By  multiplying,  we  obtain  a^ — 2ab        (jfi 2a6+62 

-f-62,  which  proves  the  theorem. 

APPLICATION. 

1.  (5_3)2:^25— 304-9=4. 

2.  (2a;— 2/)2i=4a;2— 4a:2/+2/^. 

3.  (3a;— 52;)2z=9x2_30a:^-f  25^2. 

4.  (a^— 3ea;)2=a222_6aca;2r+9c2a;2. 

80.  Theorem  III. —  The  product  of  the  sum  and  differ- 
ence  of  two  quantities,  is  equal  to  the  difference  of  their 
squares. 

Let  a  represent  one  of  the  quantities,  and  b  a  -\^b 

the  other.  a  —b 

Then,  a^bz=  their  sum,  and  a~b=  their  a^-\-  ab 

dilference.      Multiplying,    we    obtain    a^ — b^,  —  ab—b 


which  proves  the  theorem.  a^—b^ 

APPLICATION. 

•1.  (7+4)(7-4)=49-16==z33=llx3. 

2.  (2a;-f2/)(2a;-2/)=4a;2— 2/2. 

3.  (3a2+462)(3a2_452)^9a4_l664. 

4.  (3aa;+562/)(3aa;— 56?/) =9a22;2_ 2562^2. 

81.  Theorem  IV. — Any  factor  may  he  transferred  from 
one  term  of  a  fraction  to  another,  if,  at  the  same  time,  the 
sign  of  its  exponent  he  changed. 


44  RAY'S  ALGEBRA,  SECOND  BOOK. 

Take  the  fraction  -7—3.     Since  we  may  divide  both  terms  by  the 

same  quantity  without  changing  the  value  of  the  fraction,  (Ray's 
Arithmetic,  3d  Book,  Art.  136),  divide  first  by  ic^,  and  then  by  a:\ 
(Art.  70).     Thus, 

ax^     ax"^  ax^        a  a       ax^       a 

bx^  ~~   h  '  hx^  ~  bx^-^  ~  6x-2  *  *  "TT  "  bx-'' 


a        ax- 


In  a  similar  manner,  it  may  be  shown  that  t-t,  =^ — ? — . 
'  ^  bx-         b 

1      ar-2  1 

Also,  — =  =  — -  =x-^,  and  0:"'= — -,,  from  which  it  follows  that, 

Tlie  reciprocal  of  a  quantity  is  equal  to  the  same  quantify 
with  the  sign  of  its  exponent  changed. 


EXAMPLES. 

a^b          b 

S.  ^,,^=ab-r^='-; 

b'"                    a-' 

2.  a'"-   \„. 
a-'"- 

4.  a"*-":^    ^',^. 
a"-'" 

82.  Theorem  V. — Any  quantity,  whose  exponent  -is  0,  is 
equal  to  unity. 

If   we  divide  a-  by  a^,  and    apply  the  rule   for   the   exponents 

0,2 
(Art.  70),  we  find  — ^=a2-2_(;(0.    y^y^.^  since   any  quantity   is  con- 

a       ^2 
tained   in    itself   once,    — =:rl ;   therefore,  a"=^l. 

x"^  x'"' 

Similarly,  — z=^x"'-^—X^.     But-— =:1;    therefore,   x^=l,   which 

proves  the  theorem. 

By  this  notation,  we  may  preserve  the  trace  of  a  letter,  which  has 

a^b 
disappeared  in  division.     Thus,  —j-=za^-^b^-^—a^b^=.a. 

SS*  Theorem  VI. —  jThe  difference  of  the  same  power  of 
two  quantities  is  always  divisible  by  the  difference  of  the 
quantities. 


ALGEBRAIC  THEOREMS.  45 

If  we  divide  a---b~^  a^—b^;  etc.,  successively  by  a  b,  the  quo- 
tients will  be  found,  by  trial,  to  follow  a  simple  law,  both  as  to  the 
exponents  and  the  signs.     Thus, 

(^a^^b-)-^{a—b)^a+b ; 
(a^^—b"')-^{a—b)=a-^ab-\-b~; 
la^-bi)-^[a—b)=a^^a^b^ab'-irb"; 
\a^—b'')-^[CL—b)—a^^a^b-\-a-b-^ab^'-^b^^  etc. 

The  general  and  direct  proof  of  this  theorem  is  as  follows  : 
Let  us  divide  a'^—b'^  by  a—b. 

a'^^b'^la-b 


a'^^a^'-^b 


a'"-i  b—b"'=b{a'"-^ — fi'"-' ) 


.-.+*J&^),Q„„, 


In  performing  this  division,  we  see  that  the  first  term  of  the  quo- 
tient is  a'^-\  and  the  first  remainder,  6(a'"-'  — 6"*-^). 

The  remainder  consists  of  two  factors,  b  and  a"*"^ — b^-K  Now, 
if  the  second  of  these  factors,  viz.,  a"'-' — 6"'-i,  is  divisible  by  a  — 6, 
then  will  the  quantity  a'^—b*^  be  divisible  by  a—b.     That  is. 

If  the  differcjice  of  (he  same  powers  of  two  quant{tie&  is 
divisible  hy  the  difference  of  the  quantities  themselves^  then 
will  the  difference  of  the  next  higher  powers  of  the  same  quan- 
tities he  divisihlc  hy  the  difference  of  the  quantities. 

But  we  have  seen  that  a^ — b"^  is  divisible  by  a—b  ;  hence,  a^ — b^ 
is  also  divisible  by  a—b.  Again,  since  a^ — 6^  is  divisible  by  a— by 
it  follows  that  a"*— 6*  is  divisible  Tsy  it,  and  so  on;  which  proves  the 
theorem  generally. 

84.  Lemma. — In  proving  the  next  two  theorems,  it  is 
necessary  to  notice,  that  the  even  powers  of  a  negative 
quantity  are  positive,  and  the  odd  powers  negative.     Thus, 

— a,  the  1st  power  of  — a,  is  negative. 

— aX — a:=a^,  the  2d  power,  is  positive 

— aX— <^X — ^= — ^^  *^i6  3d  power,  is  negative. 

— aX~«X-"<^X— ^=^^  ^^e  4th  power,  is  positive ;  and  so  on. 


46  KAY'S  ALGEBRA,  SECOND  BOOK. 

85.  Theorem  VII. —  TJie  difference  of  the  even  powers  of 
the  same  degree  of  two  quantities^  is  always  divisible  hy  the 
sum  of  the  quantities. 

If  we  take  the  quantities  a — b  and  a"* — 6*",  and  put  —  c  instead 
of  6,  a — b  will  become  a — ( — c)=(X-|-C;  and  when  m  is  even,  b^  will 
become  c"^,  and  a^ — 6"*  will  become  a"*— (-f  c^^j^a^i — c'"  :  but 
a"* — b^  is  always  divisible  by  a — b ; 

Therefore,  a"^ — c"*  is  always  divisible  by  a-{-C  when  7)1  is  even, 
which  is  the  theorem. 

EXAMPLES. 

1.  (^a'^—b'^)^{a^b)=a—b. 

2.  \a'^—b^)^{a-Yb)=a^—a^b+ab'^—b\ 

3.  \a^—b^)^{a-]~b)=za''—a^b-]-a^b'^—a^b^-\-a¥—b\ 

86.  Theorem  VIII. —  Tlie  sum  of  the  odd  powers  of  the 
same  degree  of  two  quantities^  is  always  divisible  by  the  sum 
of  the  quantities. 

If  we  take  the  quantities  a — 6  and  a"^—b"^,  and  put  — c  instead 
of  6,  a—b  will  become  a — (— c)=a-f  c;  and  when  m  is  odd,  6"*  will 
become  —c"",  (Art.  84),  and  a'^—b'^  will  become  a"*— (— c'") 
=:a"^-\-d^:  but  am—b"^  is  always  divisible  by  a—b\ 

Therefore,  a^-\-d^  is  always  divisible  by  a+C  when  m  is  odd, 
which  is  the  theorem. 

EXAMPLES. 
1.  (a3+63)_^(ct4-6)=ra2— a6-f  62. 

3.  \a^j^b~)-^{a-]rb)=af>—a^b-\-a^b'^~a^ty^^a^b^—ab^-\-b^. 

By  a  method  of  proof  similar  to  that  employed  in  Theorem  VI., 
it  may  be  shown  that  the  sum  of  two  quantities  of  the  same  degree 
can  never  be  divided  by  the  difference  of  the  quantities.  Thus, 
a+6,  a2_|.ft2  a3-|-6^,  a^-\-b^,  etc.,  are  not  divisible  by  a — b. 

When,  in  either  of  the  last  three  theorems,  a  or  6  becomes  unity, 
the  form  of  the  quotient  will  be  obvious.     Thus, 

(«••■»— l)-^(a-l)=aH«^+a24-a+l. 
(1-f aS)-=-(l-fa}=:l— a-f a2— a3-f «S  etc. 


FACTORING.  47 


FACTORING. 

8T-  The  following  summary  of  the  principles  of  arith- 
metic should  be  remembered : 

Proposition  I. — A  factor  of  any  number  is  a  factor  of 
any  multiple  of  that  number. 

Proposition  11. — A  factor  of  two  numbers  is  a  factor  of 
their  sum. 

From  these  are  inferred  the  following,  and  the  converse 
of  each : 

1.  Every  number  ending  in  0,  2,  4,  6,  or  8,  is  divisible 
by  2. 

2.  Every  number  is  divisible  by  4,  when  the  number 
denoted  by  its  two  right  hand  digits  is  divisible  by  4. 

3.  Every  number  ending  in  0  or  5,  is  divisible  by  5. 

4.  Every  number  ending  with  0,  00,  etc.,  is  divisible 
by  10,  100,  etc. 

88.  A  Divisor  or  Factor  of  a  quantity,  is  a  quantity 
that  will  exactly  divide  it  without  a  remainder.  Thus, 
a  is  a  factor  or  divisor  of  a6,  and  a-\-x  is  a  divisor  or  fac- 
tor of  a^ — x^ 

89.  A  Prime  Quantity  is  one  which  is  exactly  divisible, 
only  by  itself  and  unity.  Thus,  x,  y,  and  x-{-z,  are  prime 
quantities ;  while  xy,  and  ax-\-az,  are  not  prime. 

90.  Two  quantities  are  said  to  be  prime  to  each  other, 
or  relatively  prime,  when  no  quantity  except  unity  will 
exactly  divide  them  both.  Thus,  ab  and  cd  are  prime  to 
each  other. 

Ol.  A  Composite  Quantity  is  one  which  is  the  prod- 
uct of  two  or  more  factors,  neither  of  which  is  unity. 
Thus,  a^ — x"^  is  a  composite  quantity,  the  factors  being 
a-{-x  and  a — x. 


48  RAY'S  ALGEBRA.  SECOND  BOOK. 

02.  To  separate  a  monomial  into  its  prime  factors, 

Rule. — Resolve  the  coefficient  into  its  prime  factors;  then, 
these  with  the  literal  factors  of  the  monomials,  will  he  the 
prime  factors  of  the  given  quantity. 

1.  Find  the  prime  factors  of  ISah^.    Ans.  2x3x3x<^-?>-&. 

2.  Of  28.x V Ans.  2X^X1  XX.X.1/.Z.Z.Z, 

3.  Of  210ax'i/z\      .     ,      Ans.  2x^X^X1 -a.x.x.x.i/.z.z. 

03.  To  separate  a  polynomial  into  its  factors,  when  one 
of  them  is  a  monomial, 

Kule. — Divide  the  given  quantify  hy  the  greatest  monomial 
(hat  will  exactly  divide  each  of  its  terms.  The  divisor  will 
be  one  factor,  and  the  quotient  the  other, 

1.  Separate  into  factors,  a-\-ax.     .     .    Ans.  a(l-{-ic). 

2.  xz-\-yz Ans.  z(x-\-yy 

3.  x^y-\-xy^ Ans.  xy{x-\-y'). 

4.  ^aV-^^a^hc Ans.  3a6(26+3ac.) 

5.  a^bx^y — ab'^xy'^-\-abcxyz^.  Ans.  abxy(ax'^ — hy-\-cz^'). 

04«  To  separate  any  binomial  or  trinomial  which  is  the 
product  of  two  or  more  polynomials,  into  its  prime  factors. 

1st.  Any  trinomial  can  be  separated  into  two  binomial  factors, 
when  the  extremes  are  squares  and  positive,  and  the  middle  term  is 
twice  the  product  of  the  square  roots  of  the  extreme  terms. 

The  factors  will  be  the  sum  or  difference  of  the  square  roots  of 
the  extreme  terms,  according  as  the  sign  of  the  middle  term  is  plus 
or  minus.     (See  Arts.  78,  79.) 

Thus,  a^+2a6+62z=(a4-6)(a+6); 
a^—2ab+b^={a—b){a—b). 

2d.  Any  binomial,  which  is  the  diflFerence  of  two  squares,  can  be 
separated  into  factors,  one  of  which  is  the  sum  and  the  other  the 
diflFerence  of  their  roots.     (See  Art.  80.) 

Thus,  a^—b-=(a-]-b){a—b). 


FACTORING. 


49 


3d.  Any  binomial  which  is  the  difFerenc^^  of  the  same  powers  of 
two  quantities,  can  be  separated  into  at  least  two  factors,  one  of 
which  is  the  difference  of  the  two  quantities.     (See  Art.  83.) 


x^—xf^{x—y){x^-\-xy-^y^). 


Thus, 

Similarly,    x^ — i/^^=(x — y){x'^-{--x,^y-\-^'^y'^-{-xy^-\-y^). 

4th.  Any  binomial  which  is  the  difference  of  the  even  powers  of  two 
quantities,  higher  than  the  second  degree,  can  be  separated  into  at 
least  three  factors.     (See  Art.  85.) 

Thus,    a4— 64=(a2+62)(a2— 62)=(a2^62)(«^6)(a-6). 

5th.  Any  binomial  which  is  the  sum  of  the  odd  powers  of  two 
quantities,  can  be  separated  into  at  least  two  factors,  one  of 
which  is  the  sum  of  the  quantities.     (Art.  86.) 

Thus,   a3+63=(a+6)(a2— a6+62j 

6th.  The  following  examples  of  the  factoring  of  binomials  com- 
posed of  the  sum  of  like  even  powers  of  quantities  may  be  verified 
either  by  multiplication  or  by  (iivision  : 

a2+62^(a+v/'2^+6)(a— 1/2^+6). 
a4+64=(a2+i/2.o6+62)(a2— v^2.a6+62). 

a6+66^(a2+62)(a4_«262_^64)^(«2_|_J2)(^J^_^3,„^,_|_52) 

(a2_v/3:a6+62). 

a8+68==(a*+l/2:a262-f&*)(a*-V^-a'6'+6*). 
aio+6io=(a24-&2)(a8_a662_|,a464— a266-f68). 

al  2_|-61  2=..(a4_|.64)(tt8_a464_^58). 

mm  mm 

a-^rnJ^h^^={a''^V2.a  ^  b^  +6'")(a'»— V  2^a^  6  2"  -f6"*). 
a3'«+63'»=(a'"+6'")(a2"»— a'"6'«-|-62"'). 

Separate  the  following  into  their  simplest  factors; 


1.  c^-\-2cd+d^. 

2.  a^x^-{-2axhj+y^. 

3.  2bx^y^-\-20xyH-riz\ 

4.  9Z^— 6z222_j.^4, 

5.  4)712x2 — imn^x-\-n^. 

6.  x^—z\ 

7.  9a2x*— 25. 
2d  Bk.        5^^ 


8.  U—a^b^z\ 

9.  a^—xK 

10.  2^4-1. 

11.  t/3-1. 

12.  a^x^—b^yK 

13.  x^-jry^. 

14.  x^ — y^. 


50 


RAY  S  ALGEBRA,  SECOND  BOOK. 


94:.  To  separate  a  quadratic  trinomial  into  its  factors. 

A  Quadratic  Trinomial  is  of  the  form  x^-\-ax-\-b,  in 
which  the  sign  of  the  second  term  may  be  either  plus  or 
minus. 

Such  a  quantity  may  be  resolved  into  factors  by  inspection.  Ob- 
serve carefully  the  product  resulting  from  the  multiplication  of  two 
factors  of  the  form  x-\-a,  and  a;+6.  Thus,  a;2— 5a;+6=:(a:— 2)(a:— 3), 
since  the  first  term  of  each  factor  must  be  X,  and  the  other  terms, 
— 2  and  — 3,  must  be  such  that  their  sum  will  be  — 5,  and  their 
product  -j-6. 

Trinomials  to  be  decomposed  into  binomial  factors. 

1.  .'c2+3x+2 Ans.  (x-\-l)(x-{-2) 

Ans.  (x — 3)  (a: — 5) 


x 


'    SxJrU. 
x-^—x—2. . 
x-^-\-x—12. 
x'—x-12. 

x^j^2x—Zh 


Ans.  (ic4-l)(x— 2) 
Ans.  (a^— 3)(x+4) 
Ans.  (x+3)(.x— 4) 
Ans.  (x— 5)(.x+7) 

95.  Examples  to  be  resolved  into  factors,  by  first  sep- 
arating the  monomial  factor,  and  then  applying  Arts.  93 
and  94. 

Ex.  1.  a7?y — ax}^^axrj{x'^—y'^)=^axy(x-\-y)(x—y). 


^ax'^-\-Qaxy-\-^ay^. 
2cx'—\2cx^\^c. 
Ziii^n — 3??i?i'.  . 
2d(?y — 2xif'.     .     . 
2x'-^6x—S.  .     . 


.  .  .  Ans.  Sa(x-\-y){x^y). 
.  .  .Ans.  2<a;— 3)(a:— 3). 
.Ans.  3wi»(m-j-7j)(m — ?i). 
Ans.  2xy(x'-\-y')ix-ry)(x~y). 
.  .  .  Ans.  2(.'r+4)(x— 1) 
.     .      Ans.  2x(x-{-1Xx—by 


2x'+4x'—l0x.  . 

Solve  the  following,  by  first  indicating  the  operations  to 
be  performed,  and  then  canceling  common  factors. 

8.   Multiply  4x — 12  by  1— a;^  and  divide  the  product 
by  2+ 2a;. 

(4a;-12)(l-a;2)     4(a;-3)(l-fa:)(l    ■-^)_9,^_a  vi  _,.^^- 
24-2a;         ~  2(l+a;)  ^        ^^        ^ 

2-^ix  -3-x2).^&c— 6-2x2. 


GREATEST  COMMON  DIVISOR.  51 

9.  Multiply  x^A^1xy-\-y'^  by  x — y^  and  divide  the  prod- 
uct by  x^ — y"^.  Ans.  x-^y 

10.  Multiply  together  1 — c,  1 — c^,  and  1-f  c^  and  divide 
the  product  by  1 — 2c-\-&.  Ans.  l-j-c-j-c^-j-c'. 

11.  Multiply  x'^—x^—^^x  by  x^+llx-j-SO,  and  divide 
the  product  by  the  product  of  a;^— 36  and  x^4-10x4-25. 

Ans.  X. 


GREATEST    COMMON    DIVISOR. 

96.  A  Common  Divisor,  or  Common  Measure,  is  any 

quantity  that  will  exactly  divide  two  or  more  quantities. 
Thus,  ah  is  a  common  divisor  of  a6^  and  ahx. 

Remark. — Two  quantities  often  have  more  than  one  common 
divisor.  Thus,  a^cx  and  abdx  have  three  common  divisors,  a,  a:, 
and  ax. 

97.  The  Greatest  Common  Divisor,  or  Greatest  Com- 
mon Measure  of  two  quantities,  is  the  greatest  quantity 
that  will  exactly  divide  each  of  them.  Thus,  the  greatest 
common  divisor  of  Qa?hx^  and  ^a?cxz  is  Za^x. 

98.  Quantities  that  have  a  common  divisor  are  said  to 
be  commensurable;  and  those  that  have  no  common  divisor, 
incommensurable. 

Note. — G.C.D.  stands  for  greatest  common  divisor. 

99.  To  find  the  G.C.D.  of  two  or  more  monomials. 

1.  Let  it  be  required  to  find  the  G.C.D.  of  the  two  mono- 
mials, 14a'c.x  and  2\a^hx. 

By  separating  each  quantity  into  its  prime  factors,  we  have 
\^^cx=ily2y^aaacx,  and  21a^bx=.ly^^yaabx. 


52  RAY'S  ALGEBRA,  SECOND  BOOK. 

The  only  factors  common  to  both  these  quantities,  arc  7,  aa  or  a^, 
and  X\  hence,  both  can  be  divided  by  either  of  these  factors,  or  by 
their  product,  ICL^X^  and  by  no  other  quantity ;  therefore,  la^x  is 
their  G.C.D.     Hence. 


TO  FIND  THE  GREATEST  COMMON  DIVISOR  OF  TWO  OR  MORE 
MONOMIALS, 

Kule. — 1.  Resolve  the  quantities  into  their  prime  factors. 
2.   Multiply  together  those  factors  that  are  common  to  all 
the  terms,  for  the  greatest  common  divisor. 

2.  Find  the  G.C.D.  of  ^a'xy,  9a'a;^  and  Iha'xhf. 

OPERATION.  Here,  3  is  the  only  numerical  factor, 

Qa^xy    —Sx2a^xy  and  a  and  X  the  only  letters  common  to 

9a%3     =3><8a3^3  all  the  quantities.     The  least  powers  of 

Iba^x^y^  ^SX^Ci'^^^y^      «  and  x,  are  a^  andiC;  hence,  the  G.C.D. 

is  3a-x. 

Find  the  G.C.D.  of  the  following  quantities : 

8.  lbahc\  and  21h'cd Ans.  Sbc. 

4.  4a^b,  lOtt-'c,  and  l^a^'hc Ans.  2a\ 

6.  4:axy,  20xyz,  and  12xyz\    .     .     .     Ans.  4^;^^ 
6.   12aV2^,  18aa;V,  SOd^x^z,  and  6ax^zK     Ans.  6ax^z. 

lOO.  Previous  to  investigating  the  rule  for  finding  the 
G.C.D.  of  two  polynomials,  it  is  necessary  to  introduce  the 
following  propositions : 

Proposition  I. — A  divisor  of  any  quantity  is  also  a  divisor 
of  any  midtiple  of  that  quantity. 

Thus,  if  A  will  divide  B,  it  will  divide  2B,  3B,  etc. 

Proposition  II. — A  divisor  of  two  quantities  is  also  a  divi- 
sor of  their  sum  or  their  difference. 

Thus,  if  A  will  divide  B  and  C,  it  will  divide  B-f  C,  or  B— C. 
ThiB  is  evident  from  Art.  74. 


GREATEST  COMMON  DIVISOR.  53 

lOl.  Let  it  be  required  to  find  the  G.C.D.  of  two  poly- 
nomials, A  and  B,  of  wliich  A  is  tlie  greater. 

If  we  divide  A  by  B,  and  there 
16  no  remainder,  B  is  evidently  the  B)A(Q 

G.C.D.,  since  it  can  have  no  divisor  BQ 


greater  than  itself.  A— BQ^R,  1st.  Rem.    . 

Divide  A  by  B,  and  call  the  quo- 
tient Q;  then  if  there  is  a  remainder  R)B(Q'' 
R,  it  is  evidently  equal  to  A— BQ.  RQ^ 
If,  now,  there  is  any  common  divisor  B— RQ^=R^    2d  Rem. 
of  A  and  B,  it  will  also  divide  BQ 

(Prop.  1st)  and  A— BQ  or  R  (Prop.  A=BQ  +R        Since  the 

2d);    or   the  common   divisor    must  B=RQ^4-R^    eiuaUoV^o 

divide  A,  B,  and  R,  and  can  not  be  product  of  the  divisor  by  tho 

greater  than  R.  quotient,  plue  the  remainder. 

Now,  if  R  will  exactly  divide  B, 
it  will  also  exactly  divide  BQ  (Prop.  1st)  and  BQ+R  (Prop.  2(1). 
Consequently,  it  will  divide  A,  since  A=BQ-)-R,  and  will  be  tho 
common  divisor  of  the  two  polynomials  A  and  B.  It  will  also  bo 
the  greatest  common  divisor,  since  no  divisor  of  A,  B,  and  R  can  bo 
greater  than  R. 

Suppose,  however,  that  when  we  undertake  to  divide  R  into  B,  to 
ascertain  if  it  will  exactly  divide  it,  we  find  that  the  quotient  is  Q^, 
with  a  remainder  R''. 

Now,  reasoning  as  before,  if  R''  exactly  divides  R,  it  will  also 
divide  RQ^  (Prop.  1st)  and  also  B  (Prop.  2d),  since  B=RQ^+R^; 
and  whatever  exactly  divides  B  and  R,  will  also  exactly  divide  A, 
since  A=BQ-|-R;  therefore,  if  R^  exactly  divides  R,  it  will  ex- 
actly divide  both  A  and  B,  and  will  be  their  common  divisor.  It 
will  also  be  the  greatest  common  divisor,  since  the  greatest  divisor 
of  R^  is  R^  itself. 

By  continuing  to  divide  the  last  divisor  by  the  last  remainder,  we 
may  apply  the  same  reasoning  to  every  successive  divisor  and  re- 
mainder; and  when  any  division  becomes  exact,  the  last  divisor  will 
be  the  greatest  common  measure  of  A  and  B. 

The  same  method  of  proof  may  be  applied  to  numbers ;  for  ex- 
ample, let  A=T=120,  and  B=r35. 

lOS.  When  a  remainder  becomes  unity,  or  does  not  contain 
the  letter  of  arrangement,  it  is  evident  that  there  is  no  common 
divisor  of  the  two  quantities. 


54  RAY'S  ALGEBRA,  SECOND  BOOK. 

103*  If  either  quantity  contains  a  factor  not  found  in  the 
other,  that  factor  may  be  canceled  without  affecting  the  common 
divisor.  Thus,  a  is  the  G.C.D.  of  ax  and  ay,  and  will  be,  if  we 
cancel  X  in  ax^  or  y  in  ay. 

1.04*   We  may  multiply  either  quantity  by  a  factor  not  found 
.  in  the  other,  without  changing  the  G.C.D.     Thus,  in  the  two  quan- 
tities, ax  and  ay,  if  we  multiply  ax  by  m,  or  ay  by  n,  the  G.C.D. 
will  still  be  a. 

103*  But  if  we  multiply  either  quantity  by  a  factor  found  in 
the  other,  we  change  the  G.C.D.  Thus,  in  the  two  quantities,  ax 
and  ay,  if  we  multiply  ay  by  x,  or  ax  by  y  the  G.C.D.  becomes  ax 
or  ay. 

lOO*  From  Art.  101,  it  is  evident  that  the  three  preceding 
articles  apply  also  to  the  successive  remainders. 

lOT-  It  is  evident  that  any  common  factor  of  two  quantities, 
must  also  be  a  factor  of  their  G.C.D.  Where  such  common  factor  is 
easily  seen,  we  may  set  it  aside,  and  find  the  G.C.D.  of  what  re- 
mains. 

Thus,  take  55a:  and  15a:.  Setting  aside  a;,  we  find  the  greatest 
common  measure  of  55  and  15  to  be  5.     Annexing  ar,  we  have  5a:. 

Remark. — The  illustrative  examples,  in  the  five  articles  above, 
are  monomials,  but  the  same  principles  obviously  apply  to  poly- 
nomials. 

We  shall  now  show  the  application  of  these  principles. 

1.  Find  the  G.C.D.  of  x^—z"  and  a:*— rrV. 

Here  the   second  quantity  contains  x^  as  a  operation. 

factor,  but  it   is  not  a  factor  of  the  first;  we  x^ — z^\x^ — z^ 

may,  therefore,   cancel  it  (Art.  103),   and  the  a? — xz"^     [^ 

second  quantity  becomes  x"^ — z^.     Then  divide  xz^ — z^ 

the  first  by  it.  or  [x — z)z'^ 

After  dividing,  we  find  that  z"^  is  a  factor  of 

the  remainder,  but  not  of  X^ — z^,  the  next  divi-  X^ — 0^  ^x — z 

dend.     We,  therefore,  cancel  it  (Art.  103),  and  a:^ — xz\x-\-z 

the  second  divisor  becomes  x — z.     Then,  divid-  xz — z^ 

ing  by  this,  we  find  there  is  no  remainder;  there-  xz — z^ 
fore,  x—z  is  the  G.C.D. 


GREATEST  COMMON  DIVISOR. 


55 


2.  Find  the  G.C.D.  of  o^^-f  a;V  and  x^—x^z\ 

The  factor  X-  is  common  to  both  quantities ; 
it  is,  therefore,  a  factor  of  the  greatest  divisor 
(Art.  107),  and  may  be  taken  out  and  reserved. 
Doing  this,  the  quantities  become  x^'-{-z^  and 
x^ — xz^.  The  second  quantity  still  contains  a 
common  factor,  X,  which  the  first  does  not; 
canceling  this,  it  becomes  x^ — 0-.  Then,  pro- 
ceeding as  in  the  first  example,  we  find  that 
X-^-Z  divides  without  a  remainder;  therefore, 
X^{x-\-z)  is  the  required  G.C.D. 


OPERATION. 

x^ — xz"^     \x 
or  (x\z)z'^ 

a;2_^2  \x-\-Z 

x^-\-xz\x — z 
— xz — z'^ 


3.  Find  the  G.C.D.  of  1  Oa^o^^— 4(71t,— Ga^,  and  bhx:'—lllx 

By  separating  the  monomial  factors,  we  find 

1 0a%2_4a2a;_6a2^2a2(  5.^2— 2a:— 3), 
and  56:c2— 116a:+66=6(5x2—lla:-)-6). 


The  factors  2a^  and  b  have  no 
common  measure,  and  hence  are 
not  factors  of  the  common  divisor. 
We  may,  therefore,  suppress  them 
(Art.  103),  and  proceed  to  find  the 
G.C.D.  of  the  remaining  quantities, 
which  is  found  to  be  x — 1. 


OPERATION. 

5a:2_l  la:+615a;2— 2a:— 3 
5a:2- 


2a:— 3     11 


—to  4- 9 

or  _9(a:-l) 

5a:2— 2a:— 3|a:— 1 
5a:2— 5a: 


& 


3a:— 3 
3a:— 3 


4.  Find    the    G.C.D.     of    ^a''—^ay^y\    and    Za^—^a''y 

In  solving  this  exam- 
ple, it  is  necessary,  in 
two  instances,  to  multi- 
ply the  dividend,  that 
the  coefficient  of  the  first 
term  may  be  divisible  by 
the  first  term  of  the  di- 
visor (Art.  104,). 


OPERATION. 

3a'>— 3a22/^02/2— ?/3|4a2._5cr2/_|_2/2 
4  ""^ 


1 2a^ — 1 2a^y  ^\ay^ — ^y"^ 
12a^—15a2y^Say^ 

3a^y-{-  ay2—iy^ 
4 


\Sa-^Sy 


12a^y\-  4a?/2— 16?/3  [over.] 


5G  RAY'S  ALGEBRA,  SECOND  BOOK. 

We    find    19?/2    is    a  12a'^y-\-  ^ay- — IG?/^    [brought  over.] 

factor   of    the    first    re-  I2a'^y—lbay-+  Sy^ 

mainder,  but  not  of  the  Iday'^ — 19y'' 

first  divisor,   and   hence  or   19y-(^a — y) 
can   not  be  a    factor  of 

the    G.C.D. ;     it    must,  4:a-—5ay-^y-\a-~y    g.c.d. 

therefore,  be  suppressed.  4a~—4ay        \4a~y 

Hence,  — ay^y"^ 

—ay^y^ 

TO     FIND    THE    GREATEST     COMMON     DIVISOR    OF    TWO 
POLYNOMIALS, 

108.  Rule. — 1.  Divide  the  greater  polynomial  hy  the 
less,  and  if  there  is  no  remainder,  the  less  quantity  will  he  the 
divisor  sought. 

2.  If  there  he  a  remainder,  divide  the  first  divisor  hy  it, 
and  continue  to  divide  the  last  divisor  hy  the  last  remainder, 
until  a  divisor  is  ohtained  which  leaves  no  remainder;  this 
ivill  he  the  G.  CD.  of  the  two  given  polynomials. 

Notes. — 1.  When  the  highest  power  of  the  Zca<fm^  letter  is  the 
same  in  both,  it  is  immaterial  which  of  the  quantities  is  made  the 
dividend. 

2.  If  both  quantities  contain  a  common  factor,  let  it  be  set  aside, 
as  forming  a  factor  of  the  common  divisor,  and  proceed  to  find  the 
G.C.D.  of  the  remaining  factors,  as  in  Ex.  2. 

3.  If  either  quantity  contains  a  factor  not  found  in  the  other,  it 
may  be  canceled  before  commencing  the  operation,  as  in  Ex.  3. 

4.  Whenever  it  is  necessary,  the  dividend  may  be  multiplied  by 
any  quantity  which  will  render  the  first  term  exactly  divisible  by 
the  first  term  of  the  divisor,  as  in  Ex.  4, 

6.  If,  in  any  case,  the  remainder  is  unity,  or  does  not  contain  the 
leading  letter,  there  is  no  common  divisor. 

6.  To  find  the  G.C.D.  of  three  or  more  quantities,  first  find  the 
G.C.D,  of  two  of  them;  then  of  that  divisor  and  one  of  the  other 
quantities,  and  so  on.  The  last  divisor  thus  found  will  be  the 
G.C.D.  sought. 

7.  Since  the  G.C.D.  of  any  two  quantities  contains  all  the  factors 
common  to  both,  it  may  often  be  found  most  easily  by  separating 
the  polynomials  into  factors.     (Arts.  U2  to  95.) 


LEAST  COMMON  MULTIPLE.  57 

Find  the  G.C.D.  in  the  following  quantities  : 

1.  5x2 — 2x — 3  and  5x2 — llx-\-6.    .     .     .      Ans.  x — 1. 

2.  9x^—4  and  9ic2— 15a:— 14 ^^s    3^_|_2. 

3.  a'^ah~12b'  and  a'-^bah-\-6h\       .     . 

4.  a* — X*  and  a^-\-a^x — ax- — x^ 

5.  x^—bx'-^lSx—d  and  x/'—2x'-\-4:X—S. 

6.  21x'—26x'-lrSx  and  6x'—x—2,      .     . 

7.  a;*-f2.c24-9andVa:'— Ilx2+15a;-f9.  Ans.  a;^— 2a:-f  3. 

8.  .-£2-^5:^+4,  a:--f  2.r— 8,  and  ic24.7a;4-12.    Ans.  a:-}-4. 

9.  26^— 10«^>2-|-8a26  and  9a'—Sab'-\-Sd'b'—9a'b. 

Ans.  a —  b. 

10.  ic*-|-^'^^*4-<^*  ^o^  ic*-f-aic' — a^x — a*.     Ans.  x'^^ax-\-(.r. 

11.  a;*— pa:^-j-(2 — l)a;'^+i^^  —  2    aiitl    ^*  —  qx^-]-(p — l)x^ 
-\-qx^p.  Ans.  ^2— 1. 


Ans. 

a+36. 

Ans. 

a^— a;^ 

Ans 

.  a:— 1. 

Ans. 

3x-2. 

LEAST    COMMON    MULTIPLE. 

109.  A  Multiple  of  a  quantity  is  any  quantity  thiit 
contains  it  exactly.  Thus,  6  is  a  multiple  of  2  or  of  3 ;  and 
ab  is  a  multiple  of  a  or  of  Z>;  also,  a(b — c)  is  a  multiple 
of  a  or  (b — c). 

no.  A  Common  Multiple  of  two  or  more  quantities, 
is  a  quantity  that  contains  either  of  them  exactly.  Thus, 
12  is  a  common  multiple  of  2  and  3 ;  and  20a:3/,  of  2x 
and  5^. 

111.  The  Least  Common  Multiple  of  two  or  more 
quantities,  is  the  least  quantity  that  will  contain  them  ex- 
actly. Thus,  6  is  the  least  common  multiple  of  2  and  3  y 
lOa;^,  of  2x  and  5y. 

Note. — L.C.M.  stands  for  least  common  multiple. 

112.  To  find  the  L.C.M.  of  two  or  more  quantities. 


58  RAY'S  ALGEBRA,  SECOND  BOOK, 

It  is  evident  that  the  L.C.M.  of  two  or  more  quantitiea 
contains  all  the  prime  /actors  of  each  of  the  quantities  oncCy 
and  does  not  contain  any  prime  factor  besides. 

Thus,  the  L.C.M.  of  ah  and  be  must  contain  the  factors 
a,  6,  c,  and  no  other  factor. 

Assuming  the  principle  above  stated,  let  us  find  the 
L.C.M.  of  mx,  nx,  and  m'^nz. 

Arranging  the  quantities  as  in  the 

OPERATION.  margin,  we  see  that  m  is  a  prime  factor 

m\mx    nx    ni^nz         common  to  two  of  them.    It  must,  there- 

r    nx      mnz         fore,  even  if  found  in  only  one  of  the 

X       X        mz         quantities,  be  a  factor  of  the  L.C.M.;  and 


I     1        1         mz         as  it  can  occur  but  once  in  the  L.C.M., 
myny^xy^inz=m~nxz         we  cancel  m  in  each  of  the  quantities 
in  which  it  is  found,  which  is  done  by 
dividing  by  it.     For  the  same  reason  we  divide  by  n  and  by  x. 

We  thus  find  that  the  L.C.M.  must  contain  the  factors  m,  n,  and  x\ 
also,  me,  otherwise  it  would  not  contain  all  the  prime  factors  found 
in  one  of  the  quantities.  Hence,  m'Xn'X.x'Xmz=m?nxz,  contains 
all  the  prime  factors  of  the  quantities  once,  and  contains  no  other 
factor ;  it  is,  therefore,  the  required  L.C.M.     Hence, 


TO    FIND  THE    LEAST    COMMON  MULTIPLE  OF  TWO  OR  MORE 
QUANTITIES, 

Rule. — 1.  Arrange  the  quantities  in  a  horizontal  line, 
divide  by  any  prime  factor  that  will  exactly  divide  two  or 
more  of  them,  and  set  the  quotients  and  the  undivided  quan- 
tities in  a  line  beneath. 

2.  Continue  dividing  as  before,  until  no  prime  factor,  ex- 
cept unity,  will  exactly  divide  two  or  more  of  the  quantities. 

3.  Multiply  the  divisors  and  the  quantities  in  the  last  line 
together,  and  the  product  will  be  the  L.C.M.  required. 

Or,  Separate  the  quantities  into  their  prime  factors ;  then, 
to  form  a  product,  1st,  take  each  factor  once;  2d,  if  any 
factor  occurs  more  than  once,  fake  it  the  greatest  number  of 
times  it  occurs  in  either  of  the  quantities. 


ALGEBRAIC   FRACTIONS.  59 

113.  Since  the  G.C.D.  of  two  quantities  contains  all  the 
factors  common  to  both,  if  we  divide  the  product  of  two 
quantities  hy  their  G.C.D.,  the  quotient  will  he  their  L.C.M. 

1.  Find  the  L.C.M.  of  ^a\  9ax^  and  24x^     Ans.  720^ 

2.  32xy,  'iQax'y,  bd'x(x—y).    Ans.  160aV/(a:— 3/) 

3.  Sx-{-6y  and  2x^—8/ Ans.  6x^—24/ 

4.  a^-\-x^  and  a^ — x^.       .     .       Ans.  a* — a^x-\-aa? — x* 

5.  x~l,  x'—l,  rr— 2,  and  x"^ — 4.     Ans.  .r,*  — 5a;2-j-4 

6.  x'—l,  x'^1,  (x—ly,  (x-]-iy,  x^—l,  and  x^-\-l 

Ans.  x^^ — X® — a:*-f  1 

7.  Sx'~llx-^6,  2.x2— 7a^+3,  and  6x'—1x-\-2.     (See 

Art.  113.)  Ans.  6a^—2bx'-\-2Bx—6. 


III.    ALGEBRAIC    FRACTIONS. 

DEFINITIONS. 

114.  Algebraic  Fractions  are  represented  in  the  same 
manner  as  common  fractions  in  arithmetic. 

The  quantity  below  the  line  is  called  the  denominator^ 
because  it  denominates,  or  shows  the  number  of  parts  into 
which  the  unit  is  divided  ;  the  quantity  above  the  line  is 
called  the  numerator,  because  it  numbers,  or  shows  how 
many  parts  are  taken. 

Thus,  in  the  fraction  -,  a  unit  is  supposed  to  be  di- 

c-\-d 

vided  into  c-\-d  equal  parts,  and  a — b  of  those  parts  are 
taken. 

113.  The  terms  proper,  improper,  simple,  compound,  and 
complex,  have  the  same  meaning  when  applied  to  alg'^braio 
fractions,  as  to  common  numerical  fractions. 


60  RAY  S  ALGEBRA,  SECOND  BOOK. 

116.  An  Entire  Algebraic  Quantity  is  one  not  ex- 
pressed under  the  form  of  a  fraction. 

IIT.  A  Mixed  Quantity  is  one  composed  of  an  entire 
quantity  and  a  fraction. 

lis.  Proposition.  —  The  value  of  a  fraction  is  not  altered, 
when  both  terms  are  multiplied  or  divided  by  the  same 
quantity. 

A  77lA 

Let  ^=Q-     Then,  will  — ^=Q-     For,  since  the  numerator  of  a 

fraction  may  always  be  considered  a  dividend,  and  the  denominator 
a  divisor,  if  we  multiply  the  numerator  or  dividend  by  any  quan- 
tity, as  m,  the  quotient  will  be  increased  m  times;  if  we  multiply 
the  denominator  or  divisor  by  m,  the  quotient  will  be  diminished  as 
much,  or  it  will  be  divided  by  m.  Therefore,  the  value  of  the  frac- 
tion is  not  changed. 

Or,  the  Proposition  may  be  proved  thus: 

--^  (Art.  81),  ^^—  =  -^=  (Art.  82),  -. 

A  similar  method  of  reasoning  may  be  applied  to  the  division  of 
the  terms  of  a  fraction. 


Case  I. — To  reduce  a  Fraction  to  its  Lowest  Terms. 
119.  From  Art.  118,  we  have  the  following 

Kule. — Divide  both  terms  of  the  fraction  by  any  quantity 
that  will  exactly  divide  them,  and  continue  this  process  as 
long  as  possible. 

Or,  Divide  both  terms  by  their  greatest  common  divisor. 

Or,  Resolve  both  terms  into  their  prime  factors^  and  then 
cancel  those  factors  which  are  common. 

In  algebraic  fractions,  the  last  is  generally  the  best  method. 


ALGEBRAIC   FRACTIONS. 


61 


1.  Reduce  jj-j — ^3  to  its  lowest  terms. 

10aex^_  2acx^  _2ax^_2a 
15bot^~3bc^~3bxS~3dx'         ' 

^     ^     ,     lOacx^      2a 

Or,  dividing  by  bcx\     j^^-.^^-^. 


Or, 

2. 
3. 

4. 

5. 
6. 

12. 


10acrc2_  2aX5ca:2  _  2a 
156c^  "  3tox5c^2  —  35^- 


Za^hx 

ax-\-x'^ 
Sbx — ex 

Sa^ — Sab' 
1-x 


^"^-  3^- 


1—x' 

a"" 


Ans. 
Ans. 
Ans, 


a-\-x 

a — b' 
1 

1-j-x 
1 


a 


n+l' 


Ans 


a* 


f_    mnp — my 
m^p-\-mp^ ' 

Q    2ax — 4ax^ 
o. 


Ans. 
.  Ans. 


n — m 

m-\-p' 

l-2x 


^ax  3 

^    ha^-\-hax          .          5a 
y.  — :r^ — ^— .        Ans. 


10   ^''+2.x-3       ^^^ 

'  x:^-\-bx-\-6'  '  xm\-2' 

^-   a^-4^-f5  ^a:'^— 5ar+5 


a — .X 
x-1 

X 


ic^H-l 


15ry^-f-35.x^H-3x+7 
27a:*+63a;-^— 12x'^— 28;c'      *     * 


Ans. 


x'—x-i-l' 


9x=*— 4x' 


The  following  examples  are  to  be  solved  by  factoring,  but  the 
process  requires  care  and  practice. 

-I  o    T.    T         x^-l-Ca-l-c^x-T-ac        .      , 

Id.  lieduce     .,     ;,    — ( — -—r-  to  its  lowest  terms. 

X  -{-{o-\-c)x-}-bc 

x--\-{a^c)x-\-ac—x^-{-ax^cx-\-aG 
Also,    x^^{b-^c)x^bc={x-\-c){x-\~b)] 


.'.  the  fraction  becomes 


(x-^c){x^a)  __x-{-a 
(x+c)  {x'^F)  ~  x^b 


,  Ans. 


^  .        ac-\-bi/-\-aTi/-\-bc 
a/-j-2bx-[-2ax-{-b/' 


Ans. 


62  RAY  S  ALGEBHA,  SECOND  BOOK. 

10.    — —. T Ans.  -, 

x'—y*'  x^—f 

it>-  T~m. -^°s.  ~-j-. 

a* — b^x^  a^ — bx 


-H    ax'^—bx'^^'^  ,  X- 

J-i-    -TT. T^-^ Ans. 


rn~\ 


a^bx — b^x?  '     '  '  b{a-\-bxy 

120.  Exercises  in  Division,  in  which  the  quotient  is 
a  fraction,  and  capable  of  being  reduced : 

1.  Divide  2aV  by  5a V6 Ans.  -^ 

56* 

2.  ax-\-x^  by  obx — ex Ans.    ,, 

ob — c 

a-'-^ab-^b^ 
a-\-b      • 


3.       a^ — b^  by  a^ — b^ Ans. 


4.       a'—b'  by  (a— by Ans.  ^!±^^H^' 

a — 6 


Case  II. — To  reduce  a  Fraction  to  an  Entire  or 
Mixed  Quantity. 

IISI.  Since  the  numerator  of  the  fraction  may  be  re- 
garded as  a  dividend,  and  the  denominator  as  the  divisor, 
this  is  merely  a  ease  of  division.     Hence, 

Kule. — Divide  the  numerator  by  the  denominator,  for  the 
entire  part.  If  there  be  a  remainder,  place  it  over  the  de- 
nominator, for  the  fractional  part,  and  reduce  it  to  its  lowest 
terms. 

1     T>    1         a^4-a^ — ax^  .  .      . 

X.  Keduce to  an  entire  or  mixed  quantity. 

a^ — ax  ^  ^ 

-     '.. — =a^x^ — 5 —a-\-x-\ ,  Ans. 


ALGEBRAIC  FRACTIONS.  63 

Reduce  the  following  to  entire  or  mixed  quantities : 

„     ax — x^  .  x^ 

A.    Ans.  X . 

a  a 

D.    — ^ — i— Ans.  a-\~h-\ r. 

a — h  a — h 

^-  i=3x ^°^'  ^+'^+r=:3x- 

-     ^-\^hx^                                                             ,             ,    "Ihx 
o-      .,     , Ans.  icH — 

x^ — bx  X — b 

^     aV — ^-\-xz — z — x-\-\  .  „  ,  z — 1 

X^ — 1  x-\-l 


Case  III. — To  reduce  a  Mixed  Quantity  to  the  form 
OF  A  Fraction. 

ISS.  This  is,  obviously,  the  reverse  of  Case  II.  Hence, 
we  have  the  following 

Rule. — 1.  Multiply  the  entire  part  by  the  denominator  of 
the  fraction. 

2.  Add  the  numerator  to  the  product^  if  the  sign  of  the 
fraction  be  plus^  or  subtract  it,  if  the  sign  be  minus. 

3.  Place  the  result  over  the  denominator. 

Before  applying  this  rule,  it  is  necessary  to  consider 

12S.  The  Signs  of  Fractions.— Each  of  the  several 
terms  of  the  numerator  and  denominator  of  a  fraction  is 
preceded  by  the  sign  plus  or  minus,  expressed  or  under- 
stood ;  and  the  fraction,  taken  as  a  whole,  is  also  preceded 
by  the  sign  plus  or  minus,  expressed  or  understood. 

Thus,  in  the  fraction ; — ,  the  sign  of  a^ig  plus;  of  6^  minus: 

'  x^y '  ^  r       )  ' 

while  the  sign  of  each  term  of  the  denominator  is  plus;   but  the 
sign  of  the  fraction,  taken  as  a  whole,  is  minus. 


64  RAY'S  ALGEBRA,  SECOND  BOOK. 

1S4.  It  is  often  convenient  to  change  the  signs  of  the 
numerator  or  denominator  of  a  fraction,  or  both. 

By  the  rule  for  the  signs,  in  Division  (Art.  69),  we  have, 

=+&;  or,  changing  the  signs  of  both  terms, z=-j-6. 

If  we  change  the  sign  of  the  jinmerator,  we  have = — b. 

If  we  change  the  sign  of  the  denominator,  we  have  — —  =—b.     Hence, 

1.  l%e  signs  of  both  terms  of  a  fraction  may  he  changed,, 
Without  altering  its  value  or  changing  its  sign,  as  a  whole. 

2.  If  the  sign  of  either  term  be  changed,  the  sign  of  the 
fraction  will  be  changed.     Hence,  also, 

3.  The  signs  of  either  term  of  a  fraction  may  be  changed, 
without  altering  its  value,  if  the  sign  of  the  fraction  be  changed 
at  the  same  time. 

Thus, — \ — — — — (— a— a;)=:a-f  cc. 

a—x  a—x  —a-lx        ^  '        ' 

And,  a —aA "— =a4 ■ — =— x. 

'         u—x  a—x  '  —a-\-x 

Applying  the  above  principles,  the  sign  of  the  fraction  may  be 
^ade  plus,  in  all  cases,  if  desired. 

Heduce  the  following  quantities  to  a  fractional  form : 
1.  2-1-1  and  2—3 Ans.   K}  and  |. 

o        ■       ,  ^^ — ^^  A         a^-\-x^ 

^.  a-\-x-\ Ans.  — - — . 

X  X 

o.  a^ — ax-\~x^ — — ; — .    Ans. 

a-\-x  a-{-x 

4.  2a-;.+(^=^'. Ans."- 

X  x' 

-              a'  .  ab 

o.  a —J Ans.  — -y. 


ALGEBRAIC  FRACTIONS.  65 

0.  a — X — - — — - ,     .     .  Ans. , 

a-j-x  a-\-x 

1.  1-^^^' Ans.  ^, 


Case  IV. — To  reduce  Fractions  of  Different  Denom 

iNATORs  TO  Equivalent  Fractions  having 

A  Common  Denominator. 

a    b 


\25» — 1.   Let  it  be  required  to  reduce  — ,  -,  and  -,  to 
a  common  denominator. 


If  we  multiply  both  terms  of  the  first  fraction  by  nr,  of  the  sec- 
ond by  Tnr,  and  of  the  third  by  mn,  we  have 


anr      bmr        ,  emn 

, ,  and    • . 

Tiinr     unnr  mnr 


As  the  terms  of  each  fraction  have  thus  been  multiplied  by  the 
same  quantity,  the  value  of  the  fractions  has  not  been  changed. 
(Art,  118.)     Hence, 


TO    REDUCE    fractions   TO   A   COMMON    DENOMINATOR, 

Rule. — Multiply  both  terms  of  each  fraction  by  the  prod- 
uct of  all  the  denominators^  except  its  own.     Or, 

1.  Multiply  each  numerator   by  the  product   of  all  the 
denominators  except  its  own.,  for  the  new  numerators. 

2.  Multiply  all  the  denominators  together  for  the  common 
denominator. 

Reduce  the  fractions  in  each  of  the  following  to  a  com- 
mon denominator: 

12,3  .^^yz     2xz    Zxy 

2.  -,   -,  and  - Ans.  — ,    — ,   — 

x'    y  z  xyz     xyz     xyz' 

3.  r-  and  - Ans.  —  and  -y. 

b  a  ab  ab' 

.         x'^A-ax       J  ox — CL^ 

.     .     .  Ans.    -7-^^  and ,, 

x^ — a^  X' — or 


4. 

and 

X — a 

a 

x^a 

2d  Bk. 

6 

66  RAY'S  ALGEBRA,  SECOND  BOOK. 

1!30.  It  frequently  happens,  that  the  denominators  of 
the  fractions  to  be  reduced  contain  a  common  factor.  In 
such  cases  the  preceding  rule  does  not  give  the  least  com- 
mon denominator. 

1X-1  •!  1  ah  c 

1,  Let  it  be  required  to  reduce  — ,  — ,  and  — ,  to  their 

-  ,      ^     .  m   mn  nr 

least  common  denominator. 

Since  the  denominators  of  these  fractions  contain  only  three  prime 
factors,  m,  n,  and  r,  it  is  evident  that  the  least  common  denomina- 
tor will  contain  these  three  factors,  and  no  others;  that  is,  it  will 
be  mnr^  the  L.C.M.  of  m,  mn,  and  nr. 

It  now  remains  to  reduce  each  fraction,  without  altering  its  value, 
to  another  whose  denominator  shall  be  mnr. 

To  effect  this,  we  must  multiply  both  terms,  of  the  first  fraction  by 
nr,  of  the  second  by  r,  and  of  the  third  by  m.  But  these  multi- 
pliers will  evidently  be  obtained  by  dividing  mnr  by  m,  m,n,  and 
nr\  that  is,  by  dividing  the  L.C.M.  of  the  given  denominators  by 
the  several  denominators.     Hence, 


TO  REDUCE  FRACTIONS  OF  DIFFERENT  DENOMINATORS  TO 

EQUIVALENT  FRACTIONS  HAVING  THE  LEAST 

COMMON  DENOMINATOR, 

Rule. — 1.  Find  the  L.C.M.  of  all  the  denominators;  this 
will  he  the  common  denominator. 

2.  Divide  the  L.C.M.  hy  the  first  of  the  given  denominators, 
and  midtiply  the  quotient  hy  the  first  of  the  given  numerators; 
the  product  will  he  the  first  of  the  required  numerators. 

3.  Proceed  thus  to  find  each  of  the  other  numerators. 

Reduce  the  fractions,  in  each  of  the  following,  to  equiv- 
alent fractions  having  the  least  common  denominator  : 

g       a       h      c  a     2hy    Sex 

6xy^  3x'  2y *  6xy^  6xy'  6xy' 

Q    _^ l_       ^  A„,    K^—h)  y(a±h)        z 

^-  a_|_6'  a—h'  a'—h'^'  a'—h'  '    d'—h'  '  a'-^h'^' 


ALGEBRAIC  FRACTIONS.  67 

.    m — n   m-\-n     mW  (m — ny    (^m,-\-ny     m^n^ 

tn-^ii   111 — n   m^ — n^  iiv — iv     m^ — iv     w* — nr 

Other  exercises  will  be  found  in  Addition  of  Fractions. 
Note. — The  two  following  Articles  may  be  of  frequent  use. 

12T.  To  reduce  an  entire  quantity  to  the  form  of  a 
fraction  having  a  given  denominator, 

Hule, — Multiply  the  entire  quantity  hy  the  given  denomina- 
tor^ and  write  the  product  over  it. 

1.  Reduce  a;  to  a  fraction  whose  denominator  is  a. 

ax 
Ans.  — . 
a 

2.  Reduce  ^az  to  a  fraction  whose  denominator  is  z^. 

2a2» 
Ans.  — -. 

z^ 

3.  Reduce  x-\-y  to  a  fraction  whose  denominator  is  x — y. 

.        x^ — y''- 

Ans.  ^. 

x—y 

l!38.  To  convert  a  fraction  to  an  equivalent  one  hav- 
ing a  given  denominator, 

Rule. — Divide  the  given  denominator  hy  the  denominator 
of  the  given  fraction^  and  midtiply  loth  terms  hy  the  quo- 
tient. 

1.  Convert  |  to  an  equivalent  fraction,  having  49  for 
its  denominator.  Ans.  |i. 

a  5 

2.  Convert  -^  and  —  to  equivalent  fractions  having  the 

denominator  9c^.  Ans.  -p^-—  and  ^ — 

3.  Convert and  — -^  to  equivalent  fractions  hav- 

oc—6  a+6  (a-\-hy    (a—hy 

ing  the  denominator  a^ — h'\  Ans.      .,     A-,  ~ — A-, 


68  RAYS  ALGEBRA,  SECOND  BOOK. 

Case  v.— ADDITION    AND    SUBTRACTION    OF    FRACTIONS. 
l!39. — 1.  Required  to  find  the  value  of  -,  -,  and  -. 

(1     (.1  (X 

Since  in  each  of  these  fractions  the  unit  is  supposed  to  be  divided 

into  d  parts,  it  is  evident  that  their  sum  will  be  expressed  by  the 

„       .       a-f6+c      „ 

iraction  , .     Hence, 

d  ' 

Rule  for  the  Addition  of  Fractions.— 1 .  Reduce  the 
fractions^  if  necessary^  to  a  common  denominator. 

2.  Add  the  numerators^  and.  write  their  sum  over  the  com- 
mon denominator. 

130. — 2.   Let  it  be  required  to  subtract  —  from  -. 

d  a 

The  unit  being,  in  each   case,  divided  into  the  same  parts,  the 

difference  will  evidently  be  expressed  by  — ^ — .     Hence, 

Rule  for  the  Subtraction  of  Fractions. — 1.  Reduce  the 

fractions,  if  necessary,  to  a  common  denonfilnator. 

2.  Subtract  the  numerator  of  the  subtrahend  from  the 
numerator  of  the  minuend,  and  write  the  remainder  over  the 
common  denominator. 


EXAMPLES  IN  ADDITION  OF  FRACTIONS. 

1.  Add  -    and  jv  together Ans.  ^j, 

2.  Add  -r  and   -  toj^ether Ans.  - — '-.—, 

h  a      ^  ah 

11  2 

3.  Add  = and  _ together.     .     .     .  Ans.  .j ^. 

*     ^-    ^    ^  ,  o  c   ,    Z>    ,    «  A         a-\-hx^-\-c:x^ 

4.  Find  the  value  of  — f-  — -j-  -.       .  Ans. 


X      x^      a?  x^ 

r     -r..    1  .1          1          o^   .      ad— he  .         a-\-hx 

5.  Find  the  value  of  -,-f   ,,    .    ,  ^.  .     .  Ans.         ,  . 

d  '    d{c-\-dx)  c-^dx 


ALGEBRAIC  FRACTIONS. 


69 


6.  Find  the  value  of  4+  1+  f.      Ans.  i!^±9^«. 
aO      ac      be  abc 


7.  Of  -4-  ^J-.    . 

8.  Of       -^      ■       ^ 


Ans. 


x'-{-f 


4(1 -1-a:)  '    4(1— ar)  '    2(H-ar^)" 


Ans. 


9.      Of-?=?+^^4-^''. 
10.     Of        -^       ■        -^ 


1 

1— a:*- 

Ans.  0. 


4a3(a+.'c)  '    4a'(a— a:)  *    2a''=(a^4-a:'^)" 


Ans. 


a'—3^' 


11.      Of 


a(a — i)(a — c)      6(6 — a){h — c)     c(c — d){c — 6)* 


ahc 


EXAMPLES  IN   SUBTRACTION   OF   FRACTIONS. 

on,  in  tlie  following,  from  the 


Subtract  the  second  fract 
first : 


-     5.^        -   3y 
1.   w-    and  —. 
la  7 


9        1  ^       1 

2.    y  and  -j~r. 

a — 0  a-f-o 


3. 


and 


P—9. 


p+q 


4.    and  - — ^. 

n  n — 1 


.  bx — Sory 

Ans.  — u — ^. 
7a 


.     .     .  Ans. 
.     .     .  Ans. 


26 


a'—h^' 
4pq 


Ans. 


1—271 

1 


A  ^-1 


6. 


and 


(2r+lX^+2)  (a:-f-l)(^-|-2)(x+3)- 


Ans. 


(2r+l)(a:+3/ 


70  RATS  ALGEBRA,  SECOND  BOOK. 

7.  -  and  ^  ,    .    ,  V- Ans 


c             c(c-j-c?x)  *  *  c-f-c/a:' 

1  3m-f2n,            1  3m— 2n  12m-t 

^'  2-3;7i— 271  ^"^    2*37714:271 •  9^^=:47/;^' 

9.   .       !^+^      .   and    ^—t±^-^.    A.  ^+^ 


(a — 6)(x — a)  (a — b)(x — 6)         *  (x — a)(x — by 


Find  the  value 


m 


in     r^£.  4771 — 371         m+37i  2n 

10.  Of  Q7i ^  — qTT ^^-1 ^-    •      •      •   ^^^'    1 

0(1 — n)       3(1 — 7i)       1 — u  1 — n 

11.  Of  ^ h-i — Aiis.  0. 

at)  ac  be 

12.  Of^±^ ^ ^=:^ Ans.l. 

..Q             1                 1                 a;+3                    .  x-\-S 


Case  VI.— MULTIPLICATION  OF  FRACTIONS. 

131. — 1.  Required  to  find  the  product  of  7  by  -^. 

a 
Here,  as  in  arithmetic,  we  take  the  part  of  ^,  which  is  expressed 
1  1    ^         a      a 

by  -^,  and  then  multiply  by  C.     Thus,  the  -^  part  of  r-  is  v-^,  and  c 

Or  thus,  J-  and  -j^=:ab-^  and  CO?-*  (Art.  81).     Multiplying,  we 
have  a&-^C(i-^=^.     Hence, 

IRiUle. — Multiply  the  numerators  together  for  a  new  numer- 
ator^ and  the  denominators  together  for  a  new  denominator 


Remarks. — 1st.  To  multiply  a  fraction  by  an  integral  quantity, 
reduce  the  latter  to  the  form  of  a  fraction,  by  writing  unity  beneath 
it  J  or,  multiply  the  numerator  by  the  integer. 


ALGEBRAIC   FRACTIONS. 


71 


2d.  If  either  of  the  factors  is  a  mixed  quantity,  reduce  it  to  an 
improper  fraction, 

3d.  When  the  numerators  and  denominators  have  common  factors^ 
let  such  factors  be  first  separated,  and  then  canceled. 


a+5 

a^—b^^  ~ioW~  {a+b){a—by^^b  ~2b{a~b)' 


2a2        (a4_6)2_2a2x(^+&)(«+&) 


Find  the  products  of  the  fractions  in  the  following : 
1.    ^  by  3^  and    —   by  ^,. 


X'   a     X 

a ,  -+-. 

a    X     a 


•^  and  2-\ 


x^y 


x—y 


1+6- 


and 


a;2-|-5x-l-4 


^ax   a} — x^    hc-{-hx 


x-\-y  '   a— &'  (y^—yf  ' 

8.   x4-l+-  by  cc— 1-1--. 

X      '^  X 

9    l^^by?^-!-^ 


.         x"^        ,   bed 

Ans.  —  and  . 

y  a 

.         a* — X* 
.  Ans.  — - — . 
a^x 


Ans. 


Ans. 
Ans 


Ans. 
Ans. 


xy^' 

4a:(a-f-ic) 

'■6y{c~x)' 

a\a-\-b') 

x—y 


Ans. 


Ans.  x^+H— ,. 
xf 


9£ 


;+2+ 


8a6* 


p — qx  p-^qx 

Ans.  rs-l-(H-|-2s)ic-l-g^a::^, 

Find  the  value 

»-'t^^)(^^')-(^')(^')■ 

atZ        be  ' 


72  RAY  S  ALGEBRA,  SECOND  BOOK. 

Case  VII.— DIVISION  of  fractions. 

13I3. — 1.  Required  to  find  the  quotient  of  -  by  -7. 

Here,  as  in  arithmetic,  the  quotient  of  y  by  -^  is -x-,andthequo- 
^.     ,     „  «  ^         .         1        c    .    a(i  ,.  . ,   ,  ,  ad 
tient  or  -j-  by  G  times  — „  or  -^,  is  -y-  diviaed  by  C,  or  -z — . 
b    •'               d       d         b                  ^    '       be 

a  c 

Or  thus,  J-  and  -^=  (Art.  81)  ab-^  and  cd-^.    Dividing,  we  have 

afx-'^     ad     „ 

— T-,  =^v— .     Hence, 

cd-  1      6c  ' 

Rule. — Invert  the  divisor^  and  proceed  as  in  multiplication 
of  fractions. 


Remark. — To  divide  a  fraction  by  an  integral  quantity,  reduce 
the  latter  to  the  form  of  a  fraction,  by  writing  unity  beneath  it ; 
or,  multiply  the  denominator  by  the  integer. 

Remarks  2  and  3,  Art.  131,  apply  equally  well  to  division  of  frac- 
tions. 

Required,  in  their  simplest  forms,  the  quotients 

1.  Of  — j Ans.  ~. 

xy         xy^  a^x 

2.  Of  — i =-— J Ans.-- 

a-\-  c      a — o  a^ — c^ 

„     „„  .r' — d^x      ax — a^  .         x'-fax^ 

3.  Of \ Ans.  ^~  . 

a^  X  (T 

5.  Of  't^fj^-^y^t ■.     Ans.  1. 

^ — y         ^ — y 

a    f\o       «* — ^  a^xAr^  K        «+^/  21        I    2\ 

6.  Of    ,     o      .    2-^    3   ^.3-     •    ^^^- {ce-^ax-^x'). 

a^ — Zflrar+j^       a^ — ar^  x 


8.  Of 


3ar 


ALGEBRAIC  FRACTIONS. 
2x 


2   •  x—1 


Ans 


73 
3 


4* 
x—1 


.0.  Of  (  x'—\  \-^lx—~\,     .     .  Ans.  x'-\-\-\-x-{-~. 

\  x*  I         \  X  I  '  X^  X 


To  REDUCE  A  Complex  Fraction  to  a  Simple  one. 

133.  This  is  merely  a  case  of  division,  in  which  the 
dividend  and  divisor  are  either  fractions  or  mixed  quan- 
tities. 

c  b  n 

Thus,  — ^  is  the  same  as  to  divide  a-\ —  by  m . 

'  n  c  r 

m 

r 


acr^hr 


ac-\-b     mr—n     ac-{-b 


X 


mr—n 


cmr—cn 


Or,  the  following  method,  obviously  true,  will  generally  be  found 
more  convenient. 

Multiply  both  terms  of  the  complex  fraction  by  the  product 
of  the  denominators,  or  by  their  L.  CM. 

acr^  br 
Thus,  in  the  above,  multiplying  by  cr,  we  have,  at  once,  -— — — —  , 

Solve  the  following  examples  by  both  methods : 


2:r— 2 


Ans. 


:~1 


a       c 

^- e  __g'  ^'''-  M{eh-fgy 
7      ^^ 

2d  Bk.  7* 


a-^\     a—\ 
CT— l"^aH-l 

a — 1     a-)-l 
a^bJr- 


Ans. 


a-h6H-- 


.  Ans. 


74  RAY'S  ALGEBRA,  SECOND  BOOK. 

Resolution  of  Fractions  into  Series. 

134*  An  Infinite  Series  consists  of  an  unlimited  num- 
ber of  terms  which  observe  the  same  law. 

The  Law  of  a  Series  is  a  relation  existing  between  its 
terms,  such  as  that  when  some  of  them  are  known  the 
others  may  be  found. 

Thus,  in  the  infinite  series  1 1 — ^ 3+)  ^^c.,  any  term  may 

be  found  by  multiplying  the  preceding  term  by . 

Any  proper  algebraic  fraction,  whose  denominator  is  a  polynomial, 
may,  by  division,  be  resolved  into  an  infinite  series. 

1 X 

1.  Convert  the  fraction  = into  an  infinite  series. 

l-x\l-\-x 


1-^x    1— 2a;-l-2a:2_2a;3+,  etc.  It  is  evident  that  the  law  of 

— 2x                                      '  this  series  is,  that  each  term, 

— 2a;— 2.t2  after  the   second,   is  equal   to 

-j-2a:2  the  preceding  term,  multiplied 

42a;24-2a:3  by  ~x. 

— 2a;3 

Resolve  the  following  fractions  into  infinite  series : 
2.    .j ^=1 — r^-j-r* — r^-j-r® — ^  etc.,  to  infinity. 


1 


-r-i-v 


l_|_^_y^_r*-fr«-f  r^— r»— r^^+j  etc. 


4.    —7-7=1 h-i 3+,  etc. 

a-\-b  a      a^      or 

135.  Miscellaneous  Propositions  in  Fractions. — The 

answer  to  some  general  question,  that  is,  the  solution  to  a 
literal  equation  (cf.  Arts.  162-165),  may  happen  to  be  a 

fraction:  e.  g.,  we  may  have  x  =  -j--  When  the  two  terms 
of  the  fraction  are  finite  numbers,  the  fraction,  being  the 
ratio  of  two  finite  numbers,  has  a  determinate  value.     But 


ALGEBRAIC   FRACTIONS.  75 

the  values  of  the  numerator  and  denominator  may  be 
changed,  by  reason  of  some  suppositions  as  to  the  values 
of  the  known  numbers  involved  in  the  question,  thus  giving 
rise  to  anomalous  results  requiring  explanation. 

130. — 1.  Thus,  suppose  a;  :=-r-  If,  while  the  denomina- 
tor remains  constant,  the  numerator  changes,  the  value  of 
the  fraction  varies  directly  with  the  numerator :  if  a  de- 
creases, the  fraction  decreases ;  if  a  becomes  0,  the  fraction 

likewise  becomes  0,  that  is,  -7-  =  0.  Also,  if  a  increases,  b 
being  constant,  the  fraction  increases ;  if  a  becomes  00,  the 
fraction  becomes  00  ;  that  is,  -j-  =  co. 

2.  If  the  denominator  changes  while  the  numerator  re- 
mains constant,  the  value  of  the  fraction  varies  inversely; 
that  is,  if  b  decreases,  the  value  of  the  fraction  iiicreases, 
and,  vice  versa,  if  b  increases,  the  value  of  the  fraction  de- 
creases ;  if  b  becomes  0,  the  fraction  becomes  00,  or  -tt  =  00 ; 
if,  on  the  other  hand,  b  becomes  00,  the  fraction  becomes 
0,  or  -  =  0. 

137.  If  the  numerator  and  denominator  are  both  ren- 
dered zero  simultaneously,  the  solution  assumes  the  form 

x  =  jr.     In  this  case,  the  unknown  number  x  is  said  to  be 

indeterminate,  inasmuch  as  it  may  evidently,  at  this  stage 
of  the  investigation,  have  any  value  whatever ;  since  the 
only  condition  imposed  upon  it  is,  that  it  shall  give,  when 
multiplied  by  zero,  a  product  equal  to  zero.     Hence,  the 

form  -rr  has  been  called  the  symbol  of  indetermination.  Nev- 
ertheless, it  may,  and  indeed  generally  does,  happen  that 
the  indetermination  is  only  apparent,  being  due  to  the  pres- 
ence, in  numerator  and  denominator,  of  a  common  factor 
which  the  particular  hypothesis  reduces  to  zero,  and  which 


76  RAY'S  ALGEBRA,  SECOND  BOOK, 

if  suppressed  before  making  the  hypothesis,  will  leave  the 
result  in  a  determinate  form.     Thus,  suppose  some  equation 

has  given  us  a;  = j-  :  if  we  make  b=a,  we  have  x  =  -?r: 

if,  however,  we   cancel  the  common  factor  a — b,  and  Vien 

(i^ 1 

make  b=a,  we  have  x^=2a.     So,  if  re  =  -^. «  '•  for  a=l, 

a  -\-a — z 

we  have  x=-7r:    but  cancelling  the  common  factor  a — 1 

2 
before  making  a=l,  gives  x  =  -^. 

These  considerations  show  that  it  is  not  safe  to  assunne  the  symbol 
■Q-  as  indicating  absolute  indetermination,  until  we  have  ascertained 
whether  the  result  has  not  been  caused,  as  in  the  examples  cited,  by 
the  presence  of  a  common  factor  which  becomes  zero  under  the  par- 
ticular supposition  imposed.  Finally,  it  should  be  remembered  that 
all  of  these  symbols,  discussed  in  this  and  the  preceding  article,  as 
well  as  some  others  of  the  same  character,  omitted  as  unsuited  to  an 
elementary  work,  are  to  be  interpreted  as  mere  abbreviations ;  other- 
wise, they  are  without  meaning. 

13S.  Theorem. — If  the  same  quantity  he  added  to  both 
terms  of  a  proper  fraction^  the  new  fraction  resulting  will  be 
greater  than  the  first;  hut  if  the  same  quantity  he  added  to 
both  terms  of  an  improper  fraction^  the  new  fraction  result- 
ing will  be  less  than  the  first. 

Let  m  represent  the  quantity  to  be  added  to  each  term 

of  any  fraction,  as  j ;  then,  the  resulting  traction  is  . 

Reducing   *   and  = to  a  common  denominator,  we  have 

^  h  b^7n  ' 

ah  \-am         ah-\-bm 

Since  the  denominators  are  the  same,  that  fraction  which  has  the 

greater  numerator  is  the  greater.     Now,  if  *   is  a  proper  fraction, 

or  if  a  is  less  than  6,  the  second  fraction  is  obviously  greater;  but 
if  it  is  improper,  and  a  greater  than  6,  the  second  is  less  than  the 
first;  which  proves  the  theorem. 


ALGEBRAIC   FRACTIONS.  77 

1.39.  Theorem. — If  tlm  same  quantity  he  subtracted  from 
hoth  terms  of  a  proper  fraction,  the  new  fraction  resulting 
will  be  less  than  the  first;  but  if  the  same  quantity  he  sub- 
tracted from,  both  terms  of  an  improper  fraction,  the  new 
fraction  resulting  will  be  greater  than  the  first. 

Let  m  represent  the  quantity  to  be  subtracted  from  each 
term  of  any  fraction,  as  y  ;  then,  the  resulting  fraction  is 


b — m 

Reducing  these  fractions  to  a  common  denominatoi',  we  have 

ab—am  ah — hin 

b'^—bm  b^—brn 

Reasoning  as  in  the  preceding  theorem,  when  the  original  frac- 
tion is  proper,  the  second  fraction  is  evidently  less  than  the  first ; 
when  improper,  it  is  greater. 

MISCELLANEOUS    EXERCISES. 

X        X — 3        X        x-\-Z        18 


1.  Prove  that 


X — 3  X  x-\-Z  X         x^ — 9* 

^-        ^a—b){a—c)  ~^  {b—a){b—c)  "^  {(•—a){c—b)        ' 

3.  Find  the  value  of  j  x-\ —^  I  —-  I  x — ^  j,  when 

x=o}i.  Ans.   9. 

.     „  _  x-\-2a   ,   x-\-2h       ,  4ab  a         o 

4.  Of  — L^-p—"       ,  when  .-^=—77.  Ans.  2. 

X — Za       X — 2b  a-\-b 

5.  Prove  that  the  sum  or  difference  of  any  two  quanti- 
ties, divided  by  their  product,  is  equal  to  the  sum  or  dif- 
ference of  their  reciprocals. 

fi    Tf         ^'+^^'  .  ^'+^'  .    __f!±^L__l 

''•  "•'  la—b){a—c)  "^  (b—a)(ib—c)  "^  (^c—a)(c—b)        ' 

prove  that  when  the  terms  are  multiplied  respectively  by 
6-f  c,  a-f-c,  and  a-\-b,  the  sum  =^0  ;  and  that  when  mul- 
tiplied respectively  by  be,  ac,  and  ah,  it  is  =M 


78  KAYS  ALGEBRA,  SECOND  BOOK. 

lY.    SIMPLE    EQUATIONS. 

DEFINITIONS   AND   ELEMENTARY    PRINCIPLES. 

140*  An  Equation  is  an  algebraic  expression,  stating 
t"he  equality  between  two  quantities.     Thus, 

X — 5=3 

is  an  equation,  stating  that  if  5  be  subtracted  from  cc,  the 
remainder  will  be  3. 

141.  Every  equation  is  composed  of  two  parts,  sepa- 
rated from  each  other  by  the  sign  of  equality. 

The  First  Member  of  an  equation  is  the  quantity  on  the 
left  of  the  sign  of  equality. 

The  Second  Member  is  the  quantity  on  the  right  of  the 
sign  of  equality. 

Each  member  of  an  equation  is  composed  of  one  or  more 
terms. 

142.  There  are  generally  two  classes  of  quantities  in 
an  equation,  the  known  and  the  nnhnown. 

The  Known  Quantities  are  represented  either  by  num- 
bers or  the  first  letters  of  the  alphabet ;  as,  a,  &,  c,  etc. 

The  Unknown  Quantities  are  represented  by  the  last 
letters  of  the  alphabet ;  as,  cc,  y^  z^  etc. 

143*  Equations  are  divided  into  degrees,  called  firsf^ 
second,  tliird,  and  so  on. 

The  Degree  of  an  equation  depends  on  the  highest  power 
of  the  unknown  quantity  which  it  contains. 

A  Simple  Equation,  or  an  cqiiation  of  the  first  degree,  is 
one  that  contains  no  power  of  the  unknown  quantity  higher 
than  the  first. 


SIMPLE   EQUATIONS.  79 

A  Quadratic  Equation,  or  an  equation  of  the  second 
degree,  is  one  in  which  the  highest  power  of  the  unknown 
quantity  is  a  square. 

Similarly,  we  have  equations  of  the  third  degree,  fourth 
degree,  and  so  on.  Those  of  the  third  degree  are  generally 
called  cubic  equations  ;  and  those  of  the  fourth  degree, 
hiquadratic  equations.     Thus, 

ax — 6==rC,  is  an  equation  of  the  1st  degree. 

x--\^2px—q^  "  "  "       2d       "        or  quadratic  equation. 

x^—px^zzq^    "  "  "       3d       "         or  cubic  " 

X^-\^ax^-\^px^q,      "  "       4tli     "         or  biquadratic      " 

x^-{^ax^-^^bx"-'^—c,  "      nth  degree. 

When  any  equation  contains  more  than  one  unknown 
quantity,  its  degree  is  equal  to  the  greatest  sum  of  the 
exponents  of  the  unknown  quantity,  in  any  of  its  terms. 

Thus,  xy-f  ax-— 5?/=C,  is  an  equation  of  the  2d  degree. 
X-y-\x^- — cx—a^  is  an  equation  of  the  3d  degree. 

144.  A  Complete  Equation  of  any  degree  is  one  that 
contains  all  the  powers  of  the  unknown  quantity,  from  O^up 
to  the  given  degree. 

An  Incomplete  Equation  is  an  equation  in  which  one 
or  more  terms  are  wanting. 

Thus,  X^-j-px-j-grzrO,  is  a  complete  equation  of  the  second  degree, 
the  term  q  being  equivalent  to  qx^\  since  x"=rl.     Art.  82. 

x3-fpx--|-gx-|-r=i0,  is  a  complete  equation  of  the  third  degree. 
ax^=zq^  is  an  incomplete  equation  of  the  second  degree. 
x3-[-^X=g,  is  an  incomplete  equation  of  the  third  degree. 

145.  An  Identical  Equation  is  one  in  which  the  two 
members  are  identical  ;  or,  one  in  which  one  of  the  r/iem- 
hers  is  the  result  of  the  operations  indicated  in  the  other. 

Thus,  ax—b=ax—b, 

8x— 3x=ox, 
(x-f-3)(x— 3)=x2 — 9,  are  identical  equations. 


80  RAY'S  ALGEBRA,  SECOND  BOOK. 

Equations  are  also  distinguished  as  mcmerical  and  literal. 

A  Numerical  Equation  is  one  in  which  all  the  known 
quantities  are  expressed  by  numbers;  as,  2.x^-\-'^x=^0x 
+15. 

A  Literal  Equation  is  one  in  which  the  known  quan- 
tities are  represented  by  letters,  or  by  letters  and  numbers ; 
as,   ax-[^h=^cx-\-d^  and  ax-\-h=^^x-\-b. 

140*  Every  equation  may  be  regarded  as  the  statement, 
in  algebraic  language,  of  a  particular  question. 

Thus,  a;— 5=9,  may  be  regarded  as  the  statement  of  the  follow- 
ing question:  To  find  a  number  from  which,  if  5  be  subtracted,  the 
remainder  shall  be  9. 

To  Solve  an  Equation  is  to  find  the  value  of  the  unknoivn 
quantity. 

An  equation  is  said  to  be  verified  when  the  value  of  the 
unknown  quantity,  being  substituted  for  it,  the  two  mem- 
bers are  rendered  equal  to  each  other. 

Thus,  in  the  equation  X — 5=9,  if  14,  which  is  the  true  value 
of  X,  be  substituted  instead  of  it, 

We  have,  14—5=9; 
Or,  9=9. 

14*7.  The  value  of  the  unknown  quantity,  in  any  equa- 
tion, is  called  the  root  of  that  equation. 


SIMPLE   EQUATIONS  CONTAINING   ONE  UNKNOWN 
QUANTITY. 

148.  All  the  rules  employed  in  the  solution  of  equa- 
tions are  founded  on  this  evident  principle  : 

If  we  perform  the  same  operation  on  two  equal  quantities^ 
the  results  will  he  equal. 

This  principle  may  be  otherwise  expressed  in  the  follow- 
ing self-evident  propositions,  or 


SIMPLE  EQUATIONS.  81 


AXIOMS. 


1.  Jf^to  two  equal  quantities^  the  same  quantity  he  added., 
the  sums  will  he  equal. 

2.  If,  from  two  equal  quantities,  (he  same  quantity  he  suh~ 
traded^  the  remainders  will  he  equal. 

3.  If  two  equal  quantities  he  multiplied  hy  the  same  quan- 
tity, the  products  will  he  equal. 

4.  If  two  equal  quantities  he  divided  hy  the  same  quantity ^ 
the  quotients  will  he  equal. 

5.  If  two  equal  quantities  he  raised,  to  the  same  power,  the 
results  will  he  equal. 

6.  If  the  same  root  of  two  equal  quantities  he  extracted, 
the  results  will  he  equal. 

140.  There  are  two  operatioDS  of  constant  use  in  the 
solution  of  equations.  These  are  Transposition,  and  Clear- 
ing an  Equation  of  Fractions^ 


TRANSPOSITION. 
ISO.  Suppose  we  have  the  equation  x — 6=c. 

By  Axiom  1,  Art.  148,  we  may  add  any  quantity  to  both  members 
of  this  equation  without  destroying  the  equality.  Adding  b  to 
both  sides, 

We  have,  X — 6-|  6=re-|-6; 

Or,  a;=c-f6,  since — 6-|-6"0. 

Comparing  this  result  with  the  original  equation,  we  find  that  it 
is  the  same  as  if  we  had  removed  the  term  6  to  the  other  side  of  the 
equation,  with  its  sign  changed. 

Again,  take  the  equation  a;-j-6=:C. 

Subtracting  b  from  both  sides,  Ax.  2,  ic-f  6 — 6=:C — 6; 
Or,  a:=c — b. 

Here  again  we  have  the  same  result  as  if  we  had  transposed  b  to 
the  other  side  with  its  sign  changed.  The  same  method  may  be 
employed  in  removing  a  term  from  the  second  member  of  the  equa- 
tion to  the  first.     Hence, 


Rule  of  Transposition. — Any  quantity  may  he  transposed 
from  one  side  of  an  equation  to  the  other ,  if  at  the  same 
time,  its  sign  he  changed. 

151.  To  Clear  an  Equation  of  Fractions. — 1.  Let  it 

be  required  to  clear  the  following  equation  of  fractions : 

ah       he 

Since,  by  Ax.  3,  Art.  148,  we  may  multiply  both  members  of  this 
equation  by  any  quantity  without  destroying  the  equality,  we  first 
multiply  by  a6,  the  denominator  of  the  first  fraction. 

This  gives,  ....     X ^—=^aoa. 

^       '  be 

Multiplying  both  members  again  by  be, 

We  have,    ....    bex—abx=ab^cd.    (1) 

Dividing  both  members  by  6,  Ax.  4,  Art.  148, 

AVe  have,    ....    ex—ax=abcd.     (2) 

If.  instead  of  multiplying  successively  by  ah  and  6c,  we  had,  at 
once,  multiplied  by  aby^bc,  or  a62c,  we  would  have  obtained  the 
form  (1)  by  one  operation.  By  multiplying  both  members  by  abe, 
the  L.C.M.  of  the  denominators,  we  would  have  obtained  the  reduced 
form  (2).  Of  these  three  methods,  the  third  is  the  most  simple. 
Hence, 

Eule  for  Clearing  an  Equation  of  Fractions.— KwcZ 

the  L.C.M.  of  all  the  denominators^  and  multiply  each  tarn 
of  the  equation  hy  it. 

Clear  the  following  equations  of  fractions : 

2.%—%=\ Ans.  4x— 3a:=12. 

3  4 

3.  ^  +  5=5 Ans.   3a:+2x=6a 

4  6 

4.  |_|+^__3i Ans.  6x-3x+2x:r.84. 


SIMPLE  EQUATIONS.  83 

5.  2x-^=='^^.      .  Ans.  20x-2rr+6==5a:— 15. 

When  a  fraction,  whose  denominator  is  to  be  removed,  is  preceded 
by  a  minus  sign,  the  signs  of  all  the  terms  in  the  numerator  must 
be  changed.  See  Art.  46,  2d.  Thus,  in  the  above  example,  wt, 
have  20x—{2x—6)=:5x—lb,  or  20a;— 2a:-t-6=^5a;— 15. 

6.  a:— '^^S—  ^.   A.12a:— 3a;+6=60— 2a;— 4. 


ax       hx 

he        ac 

X — a       X — a         2nh 


^      X     ^    ax       bx  ,0        7,  J 

7.    -T^ -f  1 =171.  .     .     Ans.  cx-4-a^x — b^x=aocm. 

ab        be        ac 

8. 


a-|-6       a — h       d^ — h'^' 

Ans.  ax — a^ — hx-\-ab — ax-^a^ — hx-\-ah=i2nh. 


SOLUTION  OF  SIMPLE  EQUATIONS  CONTAINING  ONLY  ONE 
UNKNOWN  QUANTITY. 

15S.  The  unknown  quantity  in  an  equation  may  be 
combined  with  the  known  quantities,  either  by  addition, 
subtraction,  multiplication,  or  division;  or  by  two  or  more 
of  these  different  methods. 

1.  Let  it  be  required  to  find  the  value  of  a;,  in  the  equation 

a-\-x=b, 
where  the  unknown  quantity  is  connected  by  addition. 
By  subtracting  a  from  each  side  (Art.  148),  we  have 
x=zb — a. 

2.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

X — a=6, 
where  the  unknown  quantity  is  connected  by  subtraction. 
By  adding  a  to  each  side  (Art.  148),  we  have 
x=zh-\-a. 

3.  Let  it  be  required  to  find  the  value  of  a;,  in  the  equation 

ax^:=h, 
where  the  unknown  quantity  is  connected  by  multiplication. 


84  RAY'S  ALGEBRA,  SECOND  BOOK. 

By  dividing  each  side  by  a,  we  have 
h 
a 

4.  Let  it  be  required  to  find  the  value  of  .-r,  in  the  equation 

a 
where  the  unknown  quantity  is  connected  by  division. 
By  multiplying  each  side  by  or,  we  have 

x=^hy^a=ab. 

From  the  solution  of  these  examples,  we  see  that 

When  the  nnJcnown  quantify  is  connected  hy  addition^  it  is 
to  he  separated  hy  suhtraction. 

When  connected  hy  suhtraction,  it  is  separated  hy  addition. 
When  connected  hy  multiplication,  it  is  separated  hy  division. 
When  connected  hy  division,  it  is  separated  hy  multiplication. 

5.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

„        24-2.r       ^, 

OX —  - — ^ —  ^=x-\-  o. 


Clearing  of  fractions,  21a:— (24— 2a:)=7x+56, 

Or,  21a:-24-|-2a:=7a;-l-56. 

Transposing,  2LT-f  2a:— 7a;=56+24; 

Reducing,  16a:=80; 

Dividing  by  16,  a:=^|g=5. 

In  this  solution  there  are  three  steps,  viz. :  1st.  Clearing 
the  equation  of  fractions ;  2d.  Transposition;  and  3d.  Reduc- 
ing like  terms,  and  dividing  hy  the  coefficient  of  x. 

Let  the  value  of  x  be  substituted  instead  of  x  in  the 
original  equation,  and,  if  it  is  the  true  value,  the  two 
members  will  be  equal  to  each  other.  This  is  called  veri- 
fication. 

24— 2a: 
Original  equation,     3a: _ — =a:-f  8. 

24 2v5 

Substituting  5  for  X,     3X5 _-i-^=5-f  8; 

Or,  15—2=5+8;  or,  13=13. 


SIMPLE  EQUATIONS.  85 


6.  Find  the  value  of  a:,  in  the  equation 

X ^=^4-  !-• 

ab  be 

1st  step,  abcx—cx—ac=abcd-\-ax. 
2d  step,  abcx—cx—ax=abcd^ac. 
Factoring,    {abe — c—a)  a:=ac(6d-f  1). 

3d  step,  x—~^ — ^^^—L. 

abc—G—a 


1S3.  From  the  solution  of  the  preceding  examples,  we 
derive  the  following 

Rule  for  the  Solution  of  a  Simple  Equation. — 1.  If 

necessary,  clear  the  equation  of  fractions,  and  perform  all  the 
operations  indicated. 

2.  Transpose  all  the  terms  containing  the  unknown  quan- 
tity to  one  side,  and  the  known  quantities  to  the  other. 

3.  Reduce  each  member  to  its  simplest  form. 

4.  Divide   both  sides    by   the   coefficient   of  the  unknown 
quantity. 

Find  the  value  of  the  unknown  quantity  in  the  following* 

ar^Y        2x-^  x-4. 

'•      14  21    ~^^^~    4    • 

1st  step,     18:K-f  42— 8a:+28-(-231=21a:— 84-, 
2d  step,     18a:— 8x-21a:=— 231— 42— 28— 84; 
Sd  step,     — lla;=— 385, 
a;=^35. 

8-3+2f=7f, 
7|=7t. 

8.  5(cc-f-l)— 2=8(a:+5) Ans.  x=Q. 

9.  3(a:— 2)+4=4(3— cc) Ans.  x=2. 

10.  5^3(4— a-)+4(3—2x)=0 Ans.  x=^\. 


86 


11. 


RAY'S  ALGEBRA,  SECOND  BOOK 

.     ,    Aus.  a;=:12. 
.     .   Ans.  a;=10. 


-  —  -4-7 


12-  i+l-i+H 


13.^  +  ^- 

a;       Zx 

14.  -2 ^=10+ -6"- 

,,     a;— 7i       3a;— 9    .    27— 5a; 

16.   5x-?^^  +  l=3x+i^+?. 


3 


r. 


Ans.  ic=:A. 
^  Ans,  cc=14. 
.  Ans.  x=:7-^^. 

Ans.  a;=r8. 


._    7a:+9      3a^+l      9.T-13      249— 9x    , 

17.  — ^ ^=       ^ j-^— .  Ans.:rz=9. 

18.  i(2x—10)—j\{Sx-40)=^U—l(b1—x). 

Ans.  cc=:l7. 

19.  i(4+|x)-K2x— l)=j| Ans.  x=i. 


Ans.  x=4:. 


20.  3ix{28-(|+24)}=3ix{2>+|} 

Ai 

21.  K=^-y)-A(l-3.^)=x-3'g(  5x-  ^^-^-^  ). 

Ans.  a;=rll. 

When  one  or  more  of  the  denominators  is  a  compound  quantity, 
as  in  the  two  following  examples,  it  is  generally  best  to  multiply 
all  the  terms  by  the  L.C.M.  of  the  other  denominators,  collect  the 
terms,  and  proceed  as  before. 


^^    9.r-f3  ^  3x-6     .  ,  3a:-f 22 


9 


Ans.  x=S. 


23.   ?^+^ 


ic-f2 


3.T,— 9    ^,  ,  3.r+9      .  .  . 

-12-^2|  +  ^^.    Ans.  0.^5. 

-9^ 


24.  hx-{-2x—a=Sx—2c Ans.  x 

25.  a'x-]-h^=h^x-\-a^ Ans.  x= 


b—V 


26. 


a-\-b      ' 
:^b^=a''-j-bx Ans.  x^a-\-b. 


SIMPLE  EQUATIOXS.  87 

^u.     hx       d       a       ex  .  ad 

^t' =T 7 Ans.  a;=-7- 

a        c        b        d  he 

»rt     a — h        a-\-h  .  c    ^ 

29.  5-l-^+3a6=0..     .     .     Ans.  x=e<l=^*) 
a  c  c — ad 

30.  l(x—a)—l(2x—Sb)—l(a—x)=10a-{-llb. 

Ans.  x=2^a-{-2ib. 

61.    —. \- -^ j-= .      Ans.  a:^— ^ ■ — - 

ab — ax       be — bx       ac — ax  a 


QUESTIONS  PRODUCING  SIMPLE  EQUATIONS   CONTAINING 
ONLY  ONE  UNKNOWN  QUANTITY. 

154.  The  solution  of  a  problem  by  algebra  consists  of 
two  distinct  parts  : 

1st.  Expressing  the  conditions  of  the  problem  in  algebraic 
language;  that  is,  forming  the  equation. 

2d.  Solving  the  equation;  that  is,  finding  the  value  of  the 
unknown  quantity. 

Sometimes  the  proposed  problem  furnishes  the  equation 
directly ;  and  sometimes  it  is  necessary,  from  the  condi- 
tions given,  to  deduce  others,  from  which  to  form  it.  In 
the  one  case,  the  conditions  are  said  to  be  explicit;  in  the 
other,  implied. 

It  is  impossible  to  give  a  precise  rule  by  means  of  which 
every  question  may  be  readily  stated,  in  the  form  of  an 
equation.  The  first  step  is,  to  understand  ftdly  the  nature 
of  the  question.  After  this,  the  equation  may  generally 
be  formed  by  the  following 

Rule. — Denote  the  required  quantity  by  one  of  the  final 
letters  of  the  alphabet;  tJien,  by  means  of  signs,  indicate  the 
same  operations  that  it  would  be  necessary  to  perform  with  the 
answer,  to  verify  it. 


88  RAYS  ALGEBRA,  SECOND  BOOK. 

1.  Find  two  numbers  such,  that  their  sum  shall  be  50, 
»r^d  their  diflerence  12. 

Let  a;  denote  the  least  of  the  two  required  numbers. 

!rhen  will     .     .  .T-f  12=  the  greater, 

And    ....  rc-fic-)- 12—50,  by  the  question. 

Transposing,    .  a:-|-a:=50— 12. 

Reducing,    .     .  2a:— 38. 

Dividing,      .     .  a:— 19,  the  less  number; 

And    ....  a;-f  12=19-|-12rz331,  the  greater  number. 

Verification,  31+19^50,  and  31-19=12. 

2.  What  number  is  that  whose  |  exceeds  its  I  by  6  ? 

Let  x^  the  required  number. 

X  X 

Then  will  its  1  part  be  denoted  by  -k,  and  its  -I  part,  by  '^. 


Therefore,     .     . 

Clearing,  .  . 
Reducing,  .  . 
Dividing,      .     . 

Verification, 


XX 

3-5=^- 

5a:— 3a:=90. 

2a:=90. 

a:=45,  the  number  required. 

1  of  45=15,  ^  of  45=9;  15-9=6. 


3.  A  can  perform  a  piece  of  work  in  6  days,  and  B  in 
8  days ;  in  what  time  will  both  together  finish  it  ? 

Let  X=  time  required.     Then,  since  A  can  perform  the  work  in 

X 

6  days,  he  will  perform  |  of  it  in  one  day,  and  in  X  days  ^  of  the 
work.    Reasoning  in  the  same  way  with  refeience  to  B, 

X        X 
We  have  --f-  c=  1,  the  whole  work  being  expressed  by  unity. 
D       o 

8a;-f6a;=48;  or,  14a:=48; 
And  X=Z^^  the  number  of  days. 

Or,  since  -  will  represent  the  part  of  the  work  which  both  per- 
form in  one  day,  the  equation  may  be  more  properly  stated  thus: 

6^8~a;' 


Then,   8a;-|-6a;=48,  as  before;  and  a:=3|. 


SLMPLfi  EQUATIONS.  89 

4.  Divide  |500  among  A,  B,  and  C,  so  that  B  shall 
have  $20  more  than  A,  and  C  f  75  more  than  A. 

Let    a:=A's  share;  a:-|-20=B's;  and  a:-f  75^C's. 
Then,        .     .     a:-f  a:-t-20-fa;-f  75=500,  by  the  question. 

Reducing, 3a;-f95=500. 

Subtracting  95  from  each  side,  3rc=405. 

Dividing,  a:  ==135,  As  share;  a:+20=155,  B's;  a:-f 75^210,  C's. 

Verification,  1354-155+210=500. 

5.  A  person  in  play  lost  a  fourth  of  his  money,  and 
then  won  back  $3  ;  after  which  he  lost  a  third  of  what 
he  now  had,  and  then  won  back  $2  ;  lastly,  he  lost  a 
seventh  of  what  he  then  had,  and  after  this  found  he  had 
but  $12  remaining;  what  had  he  at  first? 

Let x=:i  money  he  had  at  first. 

Then, -7=  first  loss. 

'  4 

?>x 

Subtracting  and  adding  3, -^-}-  ^^^  l^^d  after  1st  game. 

X 
\  of  the  above,  or,     .     .      -^-\-  1=  second  loss. 

X 
Subtracting  and  adding  2,   ^A^  4=  had  after  2d  game. 

X       4 

1  of  the  above,  or,      .     .     =-;+  ■=.■=  third  loss. 
7  14      7 

Zx    24 

Subtracting,     ....     ---(-"^=  had  after  3d  game. 

Zx    24 
Then, —+-y  =12;  from  which  we  find  a;=S;20. 

6.  Out  of  a  cask  of  wine  which  had  leaked  away  i,  35 
gallons  were  drawn,  and  then,  being  gauged,  it  was  -J  full ; 
how  much  did  it  hold? 

Let  a:--  the  number  of  gallons  it  held; 

Then,  f=         «  "         "         leaked  out. 

0 

X 

There  had  been  taken  away  c+^^  gallons. 
2d  Bk.  8 


90  RAY'S  ALGEBRA,  SECOND  BOOK. 

There  remained  x—  (  ^+35  \  gal.;    .-.  x—  I    f+35  W--. 
From  which  the  answer  is  readily  found. 

Y.  A  laborer  was  engaged  for  20  days.  For  each  da;y 
that  he  worked,  he  received  50  cents  and  his  boarding  ; 
and  for  each  day  that  he  was  idle,  he  paid  25  cents  for 
his  boarding.  At  the  expiration  of  the  time,  he  received 
$4  ;  how  many  days  did  he  work,  and  how  many  days  was 
he  idle? 

Let     ,     .     .     x=:  the  number  of  days  he  worked ; 

Then,      .     .     20— rr=3       "         "       "       "  was  idle. 

Also,  .     .     .     60a:z=r  wages  due  for  work. 

And   .     .     .     25(20—0:)=  the  amount  to  be  deducted  for  boarding. 

.-.  50a:— 25(20 -ic) =400. 
From  which  the  answer  is  readily  found. 

8.  What  two  numbers  are  as  3  to  5,  to  each  of  which, 
if  9  be  added,  the  sums  shall  be  to  each  other  as  6  to  7? 

Let  3a:=  the  first,  and  5a:=  the  second  number. 

Then,  3a:+9  :  5:c+9  :  :  6  :  7. 

But  in  every  proportion,  the  product  of  the  means  is  equal  to  the 
product  of  the  extremes.     (Ray's  Arith.,  3d  Book,  Art.  200.) 

Hence,  6(5a:4-9)=7(3a:-f  9). 

From  which  the  answer  is  readily  found. 

AVhen,  as  in  the  above  example,  two  or  more  unknown  quantities 
have  to  each  other  a  given  ratio, 

A&&uyne  each  of  them  a  multiple  of  some  other  unknown 
quantity^  so  that  they  shall  have  to  each  other  the  given  ratio. 

9.  A  courier,  who  traveled  at  the  rate  of  31^^  miles  in 
5  hours,  was  dispatched  from  a  certain  city ;  8  hours  after 
his  departure,  another  courier  was  sent  to  overtake  him, 
who  traveled  at  the  rate  of  22^  miles  in  3  hours.  In  what 
time  did  he  overtake  the  first,  and  at  what  distance  from 
the  place  of  departure? 

Let  X^=  the  number  of  hours  that  the  second  courier  trnvols. 
Then,  since  the  first  courier  travels  at  the  rate  of  3n   miles  iu 


SmrLE  EQUATIONS.  91 

GHx 
b  hours;  that  is,  -fg-  miles  in  1  hour,  he  will  travel  -y^  miles  in 

X  hours ;  and  since  he  started  8  hours  before  the  second  courier,  the 
whole  distance  traveled  by  him  will  be  (8-f  rr)63. 

Again,  since  the  second  courier  travels  at  the  rate  of  22^  miles 
in  3  hours,  that  is,  4^5  miles  in  1  hour,  he  will  travel  45^  miles 
in  X  hours. 

But  the  couriers  are  together  at  the  end  of  the  time  X]  therefore, 
the  distance  traveled  by  each  must  be  the  same.     Hence, 

45a: 

— -:=(8-fa:)|3|   from  which  the  answer  is  readily  found. 

10.  A  smuggler  had  a  quantity  of  brandy,  which  he  ex- 
pected would  sell  for  198  shillings  ;  after  he  had  sold  10 
gallons,  a  revenue  officer  seized  one  third  of  the  remainder, 
in  consequence  of  which,  what  he  sold  brought  him  only 
162  shillings.  Required  the  number  of  gallons  he  had, 
and  the  price  per  gallon. 

Let  x=  the  number  of  gallons  ; 

Then,  —   is  the  price  per  gallon,  in  shillings;  and  '—^ —  is  the 
X  o 

quantity  seized,  the  value  of  which  is  198 — 162=36  shillings. 

/p 20    198 

X — =36;  from  which  the  answer  is  readily  found. 


2     ^^  X 

11.  There  are  three  numbers  whose  sum  is  133 ;  the  sec- 
ond is  twice  the  first,  and  the  third  twice  the  second.  Re- 
quired the  numbers.  Ans.  19,  38,  and  76. 

12.  There  are  three  numbers  whose  sum  is  187  ;  the  sec- 
ond is  3  times,  and  the  third  4^  times,  the  first.  Required 
the  numbers.  Ans.  22,  66,  and  99. 

13.  There  are  two  numbers,  of  which  the  first  is  3|  times 
the  second,  and  their  difference  is  100.  Required  the 
numbers.  Ans.  40  and  140. 

14.  Two  numbers  are  to  each  other  as  3  to  7 ;  if  16  be 
added  to  the  first  and  subtracted  from  the  second,  the  sum 
will  be  to  the  difference  as  7  to  3.    What  are  the  numbers  ? 

Ans.  12  and  28. 


92  RAY'S  ALGEBRA,  SECOND  BOOK. 

15.  What  two  numbers  are  to  each  other  as  2  to  3,  to 
each  of  which  if  6  be  added  the  sums  will  be  as  4  to  5  ? 

Ans.  6  and  9. 

16.  A  person,  at  the  time  of  his  marriage,  was  three 
times  as  old  as  his  wife,  but  15  years  after  he  was  only 
twice  as  old.     What  were  their  ages  on  their  wedding  day? 

Ans.  Man  45,  and  wife  15. 

17.  A  bill  of  $34  was  paid  in  half  dollars  and  dimes, 
and  the  number  of  pieces  of  both  sorts  was  100  ;  how  many 
were  there  of  each  ?  Ans   60  half  dollars,  40  dimes. 

18.  There  are  three  numbers  whose  sum  is  156  ;  the  sec- 
ond is  3}  times  the  first,  and  the  third  is  equal  to  the  re- 
mainder left,  after  subtracting  the  diiference  of  the  first 
and  second  from  100.     Required  the  numbers. 

Ans.  28,  98,  and  30. 

19.  What  number  is  that,  whose  half,  third,  and  fourth 
parts,  taken  together,  are  equal  to  52?  Ans.  48. 

20.  What  number  is  that,  which  being  increased  by  its 
six  sevenths,  and  diminished  by  20,  shall  be  equal  to  45  ? 

Ans.  35. 

21.  What  nurdber  is  that,  to  which  if  its  third  and  fourth 
parts  be  added,  the  sum  will  exceed  its  sixth  part  by  51  ? 

Ans.  36. 

22.  Find  a  number  which,  being  multiplied  by  4,  be- 
comes as  much  above  40  as  it  is  now  below  it,    Ans.  16. 

23.  What  number  is  that,  to  which  if  1 6  be  added,  4 
times  the  sum  will  be  equal  to  10  times  the  number  in- 
creased by  1  ?  Ans.  9. 

24.  If  a  certain  number  be  multiplied  by  4,  and  20  be 
added  to  the  product  the  sum  will  be  32.  What  is  the 
number?  Ans.  3. 

25.  If  5  be  subtracted  from  three  fourths  of  a  certain 
number  the  remainder  will  be  equal  to  the  number  divided 
by  3.     Required  the  number.  Ans.  12. 


SIMPLE   EQUATIONS.  93 

26.  The  rent  of  an  estate  is  greater  by  8  %  than  it  was 
last  year.  The  rent  this  year  is  $1890.  What  was  it  last 
year?  Ans.  $1750. 

Observe  that  the  interest  on  any  sum  of  money  is  found  by  mul- 
tiplying the  principal  by  the  rate  per  cent.,  and  dividing  by  100. 

27.  An  estate  is  divided  as  follows :  The  eldest  child 
receives  one  fourth,  the  second  20  %,  and  the  third  15  % 
of  the  whole.  The  remainder,  which  is  $2168,  is  given  to 
the  widow.  Required  the  value  of  the  estate,  and  the  share 
of  each  child. 

Ans.  Estate  $5420;  shares  $1355,  $1084,  and  $813. 

28.  The  sum  of  two  numbers  is  30  ;  and  if  the  less  be 
subtracted  from  the  greater,  one  fourth  of  the  remainder 
will  be  3.     Required  the  numbers.  Ans.  9  and  21. 

29.  A  laborer  was  engaged  for  28  days,  upon  the  con- 
dition that  for  every  day  he  worked  he  was  to  receive  75 
cents,  and  for  every  day  he  was  absent,  he  was  to  forfeit 
25  cents.  At  the  end  of  his  time  he  received  $12.  How 
many  days  did  he  work?  Ans.  19. 

30.  At  what  time  between  two  and  three  o'clock  will  the 
hour  and  minute  hands  of  a  watch  be  together? 

Ans.  2h.  10m.  54/y  sec. 

The  face  of  a  watch  is  divided  into  60  minute  spaces,  and  the 
minute  hand  moves  twelve  times  as  fast  as  the  hour  hand. 

Let  x=  distance  from  XII  to  the  point  of  meeting;  it  will  also 
express  the  number  of  min.  after  2  when  the  hands  are  together. 
Let  X=  No.  min.  after  2  o'clock,  or  distance  min.  hand  has  gone. 
Then,  .t— 10=iz  distance  hour  hand  has  gone  after  2  o'clock. 
x^  12(a:~10)==12a;— 120; 
lla:=:120;  and  x=.\Ol^  min.,  or  the  hands  are  together 
10  min.  54-fi-  sec.  after  2  o'clock. 

31.  The  hour  and  minute  hand  of  a  clock  are  together 
at  noon  ;  when  are  they  next  together  ? 

Ans.  Ih.  5/j  min. 


94  RAY'S  ALGEBRA,  SECOND  BOOK. 

32.  At  what  time  between  8  and  9  o'clock  are  the  hour 
and  minute  hands  of  a  watch  opposite  to  each  other  ? 

Ans.  8  h.  lOlJ  min. 

33.  A  has  three  times  as  much  money  as  B,  but  if  B 
give  A  $50,  then  A  will  have  four  times  as  much  as  B. 
Find  the  money  of  each.  Ans.  A,  $750  ;  B,  $250. 

34.  From  a  bag  of  money  which  contained  a  certain 
sum,  there  was  taken  $20  more  than  its  half  j  from  the  re- 
mainder, $30  more  than  its  third  part ;  and  from  the  re- 
mainder, $40  more  than  its  fourth  part,  and  then  there  was 
nothing  left.     What  sum  did  it  contain  ?        Ans.  $290. 

35.  A  merchant  gains  the  first  year,  15  %  on  his  capi- 
tal ;  the  second  year,  20  %  on  the  capital  at  the  close  of 
the  first ;  and  the  third  year,  25  ^  on  the  capital  at  the 
close  of  the  second  ;  when  he  finds  that  he  has  cleared 
$1000.50.     Required  his  capital.  Ans.  $1380. 

36.  A  is  twice  as  old  as  B  ;  22  years  ago,  he  was  three 
times  as  old.     What  is  A's  age  ?  Ans.  88. 

37.  A  person  buys  4  houses ;  for  the  second,  he  gives 
half  as  much  again  as  for  the  first ;  for  the  third,  half  as 
much  again  as  for  the  second  ;  and  for  the  fourth,  as  much 
as  for  the  first  and  third  together  :  he  pays  $8000  for  them 
all.     Required  the  cost  of  each. 

Ans.  $1000,  $1500,  $2250,  and  $3250. 

38.  A  cistern  is  filled  in  24  minutes  by  3  pipes,  the  first 
of  which  conveys  8  gallons  more,  and  the  second  7  gallons 
less,  than  the  third  every  3  minutes.  The  cistern  holds 
1050  gallons.  How  much  flows  through  each  pipe  in  a 
minute?  Ans.  17/^,  12/^,  14l|. 

39.  A  can  do  a  piece  of  work  in  3  da3"s,  B  in  6  days, 
and  C  in  9  days.  Find  the  time  in  which  all  together  can 
perform  it.  Ans.  l-^j  days. 


SIMPLE  EQUATIONS.  95 

Let  a;=  the  required  number  of  days.     Then,  in  one  day,  A  can 
do   I,  B  1,  and  C  i,  and  all  three  -  of  the  whole  work. 

3'  b'  y  X 

Hence,    1+Hi=?- 


40.  If  A  does  a  piece  of  work  in  10  days,  whicli  A  and 
B  can  do  together  in  7  days,  how  long  would  it  take  B  to 
do  it  alone?  Ans.  23i  days. 

41.  A  performs  §  of  a  piece  of  work  in  4  days ;  he  then 
receives  the  assistance  of  B,  and  the  two  together  finish  it 
in  6  days.  Required  the  time  in  which  each  can  do  it 
alone.  Ans.  A,  14  days ;  B,  21  days. 

42.  A  person  bought  an  equal  number  of  sheep,  cows, 
and  oxen,  for  |330  ;  each  sheep  cost  $3,  each  cow  $12,  and 
each  ox  §18.     Required  the  number  of  each.     Ans.  10. 

43.  A  sum  of  money  is  to  be  divided  among  five  per- 
sons— A,  B,  C,  D,  and  E.  B  received  $10  less  than  A  ; 
C,  |16  more  than  B  ;  D,  $5  less  than  C;  E,  $15  more 
than  D  ;  and  the  shares  of  the  last  two  are  equal  to  the  sum 
of  the  shares  of  the  other  three.  Required  the  share  of 
each.     Ans.  A,  $21  ;  B,  $11 ;  C,  $27  ;  D,  $22  ;  E,  $37. 

44.  A  bought  eggs  at  18  cts.  a  dozen,  but  had  he  bought 
5  more  for  the  same  money,  they  would  have  cost  him 
2^  cts.  a  dozen  less.     How  many  did  he  buy?     Ans.  31. 

45.  A  person  bought  a  number  of  sheep  for  $94  ;  having 
lost  7  of  them,  he  sold  i  of  the  remainder  at  prime  cost, 
for  $20.     How  many  had  he  at  first?  Ans.  47. 

46.  There  are  two  places,  154  miles  distant  from  each 
other,  from  which  two  persons,  A  and  B,  set  out  at  the 
same  instant,  to  meet  on  the  road.  A  travels  at  the  rate 
of  3  mi.  in  2  hr.,  and  B  at  the  rate  of  5  mi.  in  4  hr. 
How  long,  and  how  far,  did  each  travel  before  they  met? 

Ans.  56  hr.  ;  A  traveled  84,  B,  70  mi. 

47.  A  person  bought  a  chaise,  horse,  and  harness,  for 
$450 ;  the  horse  came  to  twice  the  price  of  the  harness, 


96  RAY  S  ALGEBRA,  SECOND  BOOK. 

and  the  chaise  to  twice  the  price  of  the  horse  and  harness. 
What  was  the  cost  of  each  ? 

Ans.  Chaise  $300,  horse  $100,  harness  $50. 

48.  There  is  a  fish  whose  tail  weighs  9  lbs. ;  his  head 
weighs  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighs  as  much  as  his  head  and  his  tail.  What  is  his  whole 
weight?  Ans.  72  lbs. 

49.  Find  that  number,  which,  multiplied  by  5,  and  24 
taken  from  the  product,  the  remainder  divided  by  6,  and 
13  added  to  the  quotient,  will  still  give  the  same  number. 

Ans.  54. 

50.  In  a  bag  containing  eagles  and  dollars,  there  are 
three  times  as  many  eagles  as  dollars ;  but  if  8  eagles  and 
as  many  dollars  be  taken  away,  there  will  be  left  five  times 
as  many  eagles  as  dollars.    How  many  were  there  of  each  ? 

Ans.  48  eagles,  16  dollars. 

51.  If  10  apples  cost  a  cent,  and  25  pears  cost  2  cents, 
and  you  buy  100  apples  and  pears  for  9^  cents,  how  many 
of  each  will  you  have?         Ans.  75  apples  and  25  pears. 

52.  Suppose  that  for  every  8  sheep  a  farmer  keeps,  he 
should  plow  an  acre  of  land,  and  allow  one  acre  of  pasture 
for  every  5  sheep,  how  many  sheep  may  he  keep  on  325 
acres?  Ans.  1000. 

53.  A  person  has  just  2  hours  spare  time ;  how  far  may 
he  ride  in  a  stage  which  travels  12  miles  an  hour,  so  as  to 
return  home  in  time,  walking  back  at  the  rate  of  4  miles  an 
hour?  Ans.  6  miles. 

54.  If  65  lbs.  of  sea-water  contain  2  lbs.  of  salt,  how- 
much  fresh  water  must  be  added  to  these  65  lbs.,  in  order 
that  the  quantity  of  salt  contained  in  25  lbs.  of  the  new 
mixture  shall  be  reduced  to  i  of  a  lb.?        Ans.  135  lbs. 

55.  A  mass  of  copper  and  tin  weighs  80  lbs. ;  and  for 
every  7  lbs.  of  copper,  there  are  3  lbs  of  tin.  How  much 
copper  must  be  added  to  the  mass,  that  for  every  11  lbs. 
ol'  copper  there  may  be  4  lbs.  of  tin?  Ans.  10  lbs. 


SIMPLE  EQUATIONS.  97 

56.  A  merchant  maintained  himself  for  three  years,  at  a 
cost  of  $250  a  year ;  and  in  each  of  those  years  augmented 
that  part  of  his  stock  which  was  not  so  expended,  by  J  there- 
of At  the  end  of  the  third  year  his  original  stock  was 
doubled.     What  was  that  stock  ?  Ans.  $3700. 


SIMPLE  EQUATIONS  CONTAINING  TWO  UNKNOWN 
QUANTITIES. 

ISS.  When  an  equation  contains  two  or  more  unknown 
quantities,  the  value  of  any  one  of  them  is  entirely  depend- 
ent on  the  rest,  and  can  become  known  only  when  the  values 
of  the  rest  are  given,  or  known.     Thus,  in  the  equation 

the  value  of  x  depends  on  the  values  of  y  and  a,  and  cau 
only  become  known  when  they  are  known  ;  therefore, 

To  find  the  value  of  any  unknown  quantity,  we  must  obtain 
a  single  equation  containing  it  and  known  quantities. 

The  method' of  doing  this  is  termed  elimination,  which 
may  be  defined  briefly,  thus : 

Elimination  is  the  process  of  deducing,  from  two  or 
more  equations  containing  two  or  more  unknown  quantities, 
a  single  equation  containing  only  one  unknown  quantity. 

There  are  three  principal  methods  of  elimination : 

1st.  Elimination  hy  Suhs^fitution. 
2d.  Elimination  hy  Comparison. 
3d.   Elimination  hy  Addition  and  Siihtraction. 

150.  Elimination  by  Substitution  consists  in  finding 
the  value  of  one  of  the  unknown  quantities  in  one  of  the 
equations,  and  substituting  this  value  in  the  other  equa- 
tion. 

2a  Bk.  9* 


98  RAY'S  ALGEBRA,  SECOND  BOOK. 

To  explain  this  method,  let  it  be  required  to  find  the  values 
of  X  and  y^  in  the  following  equations: 

2a:+3?/=33,        (1) 
4a:+5?/=59.        (2) 

From  (1),  by  transposing  Zy  and  dividing  by  2,  we  have 
33-3y 

Substituting  this  value  of  a:,  instead  of  x  in  (2),  we  have 

66—6^+^=59; 

-2/=-7; 
2/=7; 

a;^3  3-3X7^6 

The  following  is  the  general  form  to  which  two  equations  of  the 
first  degree,  containing  two  unknown  quantities,  may  always  be 
reduced.  The  signs  of  the  known  quantities,  a,  6,  c,  etc.,  may  be 
either  plus  or  minus. 

axA-hyz^c,        (1) 
a'x-^b'y=&.       (2) 

From  (1),  by  transposing  6y,  and  dividing  by  a,  we  have 

a 
Substituting  this  value  of  x  in  (2),  we  have 

a'c—a'by  -f-  ab'y=a&\ 

{ah' — a'b)y=a& — a'c ; 
_  ac'—a'G 
y-ab'—a'b' 
I  a&—a'c  \ 
^_€-by_<^-f>\  ab'-a'b  } _ab'c-a'bc-ab& -\-a'bG 
jjut  ^— ^— _ —  a{ab'—a'b)    h 

b'c—bc'      „ 
=  -1.7 >j,'     Hence, 

Rule  for  Elimination  ty  Substitution. — Find  an  expres- 
sion for  the  value  of  one  of  the  unknown  quantities  in  either 


SIMPLE  EQUATIONS.  99 

equation^  and  substitute  this  value,  instead  of  the  same  un- 
known quantity,  in  the  other  equation;  there  will  thus  he 
formed  a  new  equation,  containing  only  one  unknown 
quantity. 


1.  Zx—by—  2,  \         Ans.  cc=4, 
2a:+7i/=r22.  J  y=2. 

2.  5a;— 3(a;— ?/)=13,  |    A.  x=b, 
x—y—A.  i         y=l- 


3.  4x=G^—3y,  ^        Ans.  a:=rlO, 
2a; +3^=44.  /  y=  8. 


4x-^Sy=8y+3. 


5.  ax^by=c — d,  i  _n{c—d)       _m{G--d) 


\  .  n(c—d) 

) Ans.  x=z^ ^,   y- 

)  an^brrv  ^ 


imx=ny.  )  an^brrv  ^     an-\-brrb 

Remark. — This  method  is  always  to  be  preferred  where  the 
value  of  one  of  the  unknown  quantities  may  be  found  in  terms  of 
the  other,  as  in  examples  4  and  5  above. 

157.  Elimination  by  Comparison  consists  in  finding 
the  value  of  the  same  unknown  quantity  in  two  different 
equations,  and  then  placing  these  values  equal  to  each 
other. 

To  illustrate  this  method,  we  will  take  the  same  equations  as  in 
the  preceding  article. 

2a;+3y=33,        (1) 
4a;+5y^59.        (2) 

From  (1),  by  transposing  and  dividing,  we  have  a;= — ^ — . 

KQ tyj. 

From  (2),  by  transposing  and  dividing,  we  have  a;= — ^ — . 

Placing  these  values  of  x  equal  to  each  other, 
59— 5j/_33— 31/ 
4      "~~2~' 
59—5^  =  66 — 63/,  by  clearing  of  fractions; 
y  =  7,  by  transposition. 

The  value  of  x  may  be  found  similarly,  by  first  finding  the  values 
of  y,  and  placing  them  equal  to  each  other.  Or,  it  may  generally 
be  found  most  readily  by  substitution.     Thus, 

4a;+5X7=59; 
Whence,  x=^^^^=.Q. 


100  RAY'S  ALGEBRA,  SECOND  BOOK. 

General  equations,  ax-{-by=iCj        (1) 
a'x-\-l/y=&.      (2) 

From  (1),  by  transposing  and  dividing,  Xz= -. 

Prom  (2),  by  transposing  and  dividing,  a;= ^'  . 

Equating  these  values  of  x^ 

c—hy_&—h'y 

a^c—a^by=a& — Clb^y,  by  clearing  of  fraction  ; 
{ah^—a'b)y—a&—a'G^  by  transposing; 
a& — a^G 

„        ,^,         c—ax      ^        ,„,         &—a^x 
From  (1),  y^—^-  ;     from  (2),  y^      ^,      . 

Equating  these  values  of  2/, 

&—a^xG — ax 
~~¥~""~b~' 
b&—a^bx=  b^c-ab^x ; 
{ab'—a'b)x=b'c~b&\ 

b'c-b&      „ 
iC— -XT Til-     Hence, 


Rule  for  Elimination  by  Comparison. — Find  an  expres- 
sion for  the  value  of  the  same  unknown  quantity  in  each  of 
the  given  equations^  and  place  these  values  equal  to  each  other; 
there  will  thus  he  formed  a  new  equation^  containing  only  one 
vnknown  quantity. 


1.  3a:— 2?/=  9,  ^         Ans.  ic=5, 

2.  7a;4-y=102/+7,  ^  Ans.  x^\0, 

x^y=  22/+3.  /         y^  7. 

3.  4c-f-  ^=51,  y      Ans.  x=.Q, 
%x—\^y=z  9.  J  y=Z. 


4.  ma:=7?y, )  an 

^   ^       Ans.  x= — — . 


a.  f 


x-\-y=a.  J  m-f  ri 

am 


y-= 


m-\-n 

5.  ax-\-by—p^'\  bq—dp 

cx-\-dy^q.  J        ^~bc-ad} 

aq-cp^ 

^    ad— be 


Remark.— This  method  is  generally  to  be  preferred  v^'here  the 
equations  are  literal,  and  sometimes  in  other  cases. 


SIMPLE  EQUATIONS.  101 

158.  Elimination  by  Addition  and  Subtraction  con- 
sists iu  multiplying  or  dividing  two  equations,  so  as  to 
render  the  coefficient  of  one  of  the  unknown  quantities  the 
same  in  both  ;  and  then,  by  addition  or  subtraction,  caus- 
ing the  terms  containing  it  to  disappear. 

Taking  the  same  equations  as  in  the  preceding  articles, 
2a;+3^=33,        (1) 
4xi-5y^69.        (2) 

It  is  evident  if  we  multiply  (1)  by  2,  that' the  ^coefficient  of  a;  will 
be  the  same  in  the  two  equations.  I    ;  i  ',J  I    ',  ,*  >^   'j  l^ 

4x-\-6i/=:66  (3),  by  multit)iyiog.(l),biy',2;, ,  ,.,  .^  ,  , 
4x~\-by=iz59,  (2)  brought. davvH:.  • ,    ;  ' '  \^,'  J  ','^  ]\l  ^  , 
2/:=::  7,  by  subtractiou. 

If  the  signs  of  the  coefficients  of  x  had  been  different^  the  terms 
in  X  would  have  been  canceled  by  adding. 

Having  obtained  the  value  of  y^  that  of  X  may  be  obtained  in  the 
same  way,  or  by  substitution.     Thus, 

Multiply  (1)  by  5,  and  (2)  by  3,  and  the  coefficients  of  y  will  be 
the  same  in  both. 

10a:+15i/=.lG5,  (4)  by  multiplying  (1)  by  5. 

]2a:+157/=177,  (5)  by  multiplying  (2)  by  3. 
'2xzzzl2,  by  subtracting  (4)   from  (5). 

x=  6. 

Or,  by  substitution,  from  either  of  the  original  equations.     Thus, 
From  (1)  2rc-f-3x7=33 ; 
2a:=33-21=12; 
x=6. 

General  equations,  ax-\-by=zC,  (1) 

a'x-\-b'y=^c^.       (2) 

It  is  evident  that  we  shall  render  the  coefficients  of  X  the  same  in 
both  equations,  by  multiplying  (1)  by  a^,  and  (2)  by  a. 

aa^x-\-a^by=:za^c,     (3),  by  multiplying  (1)  by  a^; 
aa^x-\-ab^y=a&,     (4),  by  multiplying  (2)  by  a; 
{ab^~a^b)y=a&—a^c,  by  subtracting; 
ac^—a^G 


102 


RAY'S  ALGEBRA,  SECOND  BOOK. 


The  coefficients  of  y  in  the  two  equations  will  evidently  become 

equal  by  multiplying  (1)  by  6'',  and  (2)  by  b.  ' 

ah^xA^bh^y—l/c,     (5)     by  multiplying  (1)  by  b^; 

a'bx^bb'y=bG%     (6)     by  multiplying  (2)  by  6; 

{ab'—a^b)x—b'G—b&^    by  subtracting; 

b'c-b&      „ 
x^=^—j~, jy.     Hence, 


Rule  for  Elimination  by  Addition  and  Subtraction. — 
1.  Multiply  or  divide  fjie  equations,  if  necessary,  so  that  one 
of  the  unknown  c^Hxntities  will  have  the  same  coefficient  in 
hoth.  _  ,    .  -- 

'  Yy^u^l^a'he^lTi^  d^ffe^^^  the  sum,  of  the  equations,  accord- 

ing as  the  signs  of  the  equal  terms  are  alike  or  unlike,  and  the 
resulting  equation  will  contain  only  one  unknown  quantity. 

Remark. — When  the  coefficients  of  the  quantity  to  be  eliminated 
are  prime  to  each  other,  multiply  each  by  the  other.  When  the 
coefficients  are  not  prime,  multiply  by  such  numbers  as  will  produce 
their  L.C.M. 

If  the  equations  have  fractional  coefficients,  they  ought  to  be 
cleared,  before  applying  the  rule. 


1.  a:+3?/.=]0,-) 
Sx-\-2y^  9.  J 

2.  2rr+3?/=18,  ^ 
Sx-2y--=  1.) 

3.  2x~  %=n 
Sx-12y=l^ 


:} 


Ans.  a:=:l, 
y=S. 

Ans.  x=zS, 

Ans.  x=zl, 
2/=-l. 


40     " 
2x—y 


a^-2/, 


+^y=h 


5.    x-{-ay=b 
ax 


—C.) 


A.  x= 


y= 


A.  x=.^, 

ac-\-b2 
a^^bi 
ab — c 
a^b' 


The  following  examples  may  be  solved  by  either  of  the 
three  methods  of  elimination  : 


1.  ^x-Ay^    8,-) 
13a;-f7y=101.J 

2.  x-:^{y-  2)^5, 
Ay-l{x^\Q)=.Z. 


Ans.  X=A, 

A.  a;=5, 
y-2. 


q    ^^y    ft 


10 


^1. 


Ans.  iC— 18, 
y-10. 


SIMPLE  EQUATIONS. 


103 


4. 


5^1/      1 


Ans.  a:=:2. 


8a;-4zz.9?/, 


;} 


6.  K^+^)+K^-2/)=59, 
6a;— 33z/=0. 


3a:+4v+3      2x+l~y  y-8 

%+5x— 8  _  x-f^     7a:4  6 
12  ~T~ 


Ans.  X- 

-6, 

y 

-4. 

Ans.  X— 

99, 

y^ 

15. 

Ans.  a:; 

=7, 

y 

-9. 

8.  ax -by 
x\y 


1} 


9.  Sax  -2by=c,    l 


10.  — [--=«, 
^3/ 


11 


be         ,          ac 
Ans.  a:—- y,  nxid  y— -.. 

Ube  ,        c/  156— a  \ 

2a-^3ab'         ^    b\  2a-f36  / 


X    y 


11.  (a2_&2)(^5a;_^3y)^(4o^_5)2a6, 

«62C 


Ans.  X- 

y 


ma—nb^ 
mb—nd' 


a-y j^-^{a-\-b^c)bx=b^y-{-{a']~2b)ab. 


Ans.  Xz 

y^ 


a^b' 

ab 

a  -b' 


Rb:m  ARK. —Transpose  b-y  \xi  (2),  multiply  by  3,  and  subtract; 
there  will  then  result  an  equation  involving  X. 


PROBLEMS   PRODUCING    SIMPLE  EQUATIONS   CONTAINING 
TWO  UNKNOWN  QUANTITIES. 

ISO.  The  questions  contained  in  Art.  154,  may  all  be 

solved  by  using  one  unknown  symbol,  although,  in  some 
cases,  there  were  two  or  more  unknown  quantities. 

It  frequently  happens,  however,  that  the  conditions  of  a 
problem  are  such  as  to  require  the  use  of  two  or  more  sym- 
bols for  the  unknown  quantities.  In  this  case,  the  number 
of  equations  must  be  equal  to  the  number  of  symbols,  and 


104  RAYS  ALGEBRA,  SECOND  BOOK. 

the  value  of  the  unknown  quantities  may  be  found  by  either 
of  the  three  methods  of  elimination. 

A  problem  may  often  be  solved  by  using  either  one  or 
more  unknown  quantities.  In  illustration,  take  the  fol- 
lowing : 

1.  The  difference  of  two  numbers  is  a,  and  the  less  is  to 
the  greater  as  m  to  n ;  required  the  numbers. 

Solution  by  using  one  unknown  quantity. 

Let  mx=z  the  less  number,  and  nx=  the  greater. 

Then,  nx—mx=a. 

a                       ma  ^  na 

.'.  mx=z ,    and  nxz 


n-^m  n—m  n — m 

Solution  by  using  two  unknown  quantities. 
Let  x=  the  less  number,  and  ?/=:  the  greater. 
Then,  y — x=a,         (1) 
And  x\y\\in\n\    or,  my—nx.        (2) 

nx 
Since  myzr^nx^  we  have  y^^- — -; 

Substituting  this  value  of  y  in  (1),  we  find  as  before, 

ma  ,  na 

X= ,    and  ?/= . 

n—m  ^     n — di 


2.  The  hour  and  minute  hands  of  a  watch  are  opposite 
at  6  o'clock  ;  when  are  they  next  opposite  ? 

Let  X^=  minute  spaces  moved  over  by  the  hour  hand,  and  2/~  min- 
ute spaces  moved  over  by  the  minute  hand.  Then,  since  the  minute 
hand  moves  12  times  as  fast  as  the  hour  hand, 

x:y'.'.l\\2,  or  y=^l2x.        (1) 
But  the  minute  hand  must  evidently  pass  over  60  minutes  moro 
than  the  hour  hand  ;  hence, 

i/^x+60.        (2) 
Substituting,  12a:=a:-f  60, 
llrc=60. 
Xz=b^~  min. 

y=i65JL  min.  =1  h.,  5A.  m. 
Hence,  the  hands  are  next  opposite  at  5^^  m.  past  7. 
In  a  similar  manner  the  period  of  coincidence  of  tlie  hands  may 
be  found. 


SIMPLE  EQUATIONS.  105 

3.  There  is  a  number  consisting  of  two  digits,  which 
divided  by  the  sum  of  its  digits,  gives  a  quotient  7  ;  but  if 
the  digits  be  written  in  an  inverse  order,  and  the  number 
thence  arising  be  divided  by  the  sum  of  the  digits  -{-4,  the 
quotient  =3.    Required  the  number.  Ans.  84. 

In  solving  questions  of  this  kind,  observe  that  any  number  con- 
sisting of  two  places  of  figures,  is  equal  to  10  times  the  figure  in  the 
ten's  place  plus  the  figure  in  the  unit's  place.  Thus,  35  is  equal 
to  10x3+5;  456^100X4+10X54  6. 

Let  a:=  the  tens'  digit,  and  ^=  the  units'  digit. 

Then,  10x+2/=  the  number. 

And  102/+a:i=:  the  number  when  the  digits  are  reversed. 

Also,  ^.^^^1.  i^,=3. 

From  these  equations  we  readily  find  a:=:8,  and  2/=4. 

4.  A  farmer  sells  to  one  man  5  sheep  and  7  cows  for 
fill,  and  to  another,  at  the  same  rate,  7  sheep  and  5  cows 
for  $93.     Required  the  price  of  a  sheep  and  of  a  cow. 

Ans.  Sheep,  $4;  cow,  $13. 

5.  If  7  lbs.  of  tea  and  9  lbs.  of  coffee  cost  $5.20,  and, 
at  the  same  rate,  4  lbs.  of  tea  and  11  lbs.  of  coffee  cost 
$8.85,  what  is  the  price  of  a  lb.  of  each? 

Ans.  Tea,  55c. ;  coffee,  15c. 

G.  A  and  B  are  in  trade  together  with  different  sums  ; 
if  $50  be  added  to  A's  money,  and  $20  be  taken  from  B's, 
they  will  have  the  same  sum;  but  if  A's  money  were  3  times, 
and  B's  5  times  as  great  as  each  really  is,  they  would  to- 
gether have  $2350.     How  much  has  each  ? 

Ans.  A,  $250;  B,  $320. 

7.  A  and  B  together  have  $9800  ;  A  invests  the  sixth- 
part  of  his  money  in  business,  and  B  the  fifth  part,  and 
then  each  has  the  same  sum  remaining.  How  much  has 
each?  Ans.  A,  $4800  ;  B,  $5000. 

Suggestion. — Let  6a;==  A's  money,  and  by^^  B's. 


lOG  RAY  S  ALGEBRA,  SECOND  BOOK. 

8.  Find  a  fraction,  such  that  if  the  numerator  and  de- 
nominator be  each  increased  by  1,  the  value  is  l;  but  if 
Bach  be  diminished  by  1,  the  value  is  |.  Ans.  ^. 

9.  Find  two  numbers,  such  that  ^  of  the  first  exceeds 
I  of  the  second  by  3,  and  |  of  the  first  and  i  of  the  sec- 
ond are  together  equal  to  10.  Ans.  24  and  20. 

^  ^  10.  A  grocer  knows  neither  the  weight  nor  the  first  cost 
of  a  box  of  tea  he  had  purchased.  He  only  recollects  that 
if  he  had  sold  the  whole  at  30  cts.  per  lb.,  he  would  have 
gained  $1,  but  if  he  had  sold  it  at  22  cts.  per  lb.,  he  would 
have  lost  $3.  Required  the  number  of  lbs.  in  the  box, 
and  the  first  cost  per  lb.  Ans.  50  lbs.  at  28  cts. 

11.  The  rent  of  a  field  is  a  certain  fixed  number  of  bu. 
of  wheat,  and  a  fixed  number  of  bu.  of  corn.  AVhen  wheat 
is  55  cts.,  and  corn  33  cts.  per  bu.,  the  portions  of  rent  by 
wheat  and  corn  are  equal ;  but  when  wheat  is  65  cts.  and 
corn  41  cts.,  the  rent  is  increased  by  $1.40.  What  is  the 
grain  rent?  Ans.  6  bu.  of  wheat,  10  of  corn. 

12.  The  quantity  of  water  which  flows  from  an  orifice  is 
proportional  to  the  area  of  the  orifice,  and  the  velocity  of 
the  water.  Now,  there  are  two  orifices,  the  areas  being 
as  5  to  13,  and  the  velocities  as  8  to  7;  and  from  one  there 
issued  in  a  certain  time  561  cubic  feet  more  than  from  the 
other.     How  much  water  did  each  discharge  ? 

Ans.  440  and  1001  cubic  feet. 

13.  Find  two  numbers  in  the  ratio  of  5  to  7,  to  which 
two  other  required  numbers,  in  the  ratio  of  3  to  5,  being 
respectively  added,  the  sums  shall  be  in  the  ratio  of  9  to 
13  ;  and  the  difibrence  of  those  sums  :=16. 

Ans.   30  and  42,  6  \nd  10. 

*  14.  A  boy  spends  30  cts.  in  apples  and  pears,  buying 
his  apples  at  4  and  his  pears  at  5  for  a  ct. ;  he  then  finds 
that  half  his  apples  and  ]  of  his  pears  cost  13  cts.  How 
many  of  each  did  he  buy  ?         Ans.  72  apples,  GO  pears- 


SIMPLE  EQUATIONS.  107 

15.  A  farmer  rents  a  farm  for  $245  per  year ;  the  till- 
able land  being  valued  at  $2  per  acre,  and  the  pasture  at 
$1.40  ;  now  the  number  of  acres  of  tillable,  is  to  half  the 
excess  of  the  tillable  above  the  pasture,  as  28  to  9.  How 
many  acres  are  there  of  each? 

Ans.  98  acres  tillable,  35  of  pasture. 

16.  Find  that  number  of  2  figures  to  which,  if  the  num- 
ber formed  by  changing  the  places  of  the  digits  be  added, 
the  sura  is  121 ;  and  if  the  less  of  the  same  two  numbers 
be  taken  from  the  greater,  the  remainder  is  9.    Ans.  65  or  56. 

lY.  To  determine  three  numbers  such  that  if  6  be  added 
to  the  first  and  second,  the  sums  will  be  in  the  ratio  of  2  to 
3 ;  if  5  be  added  to  the  first  and  third,  the  sums  will  be  in 
the  ratio  of  7  to  11  ;  but  if  36  be  subtracted  from  the 
second  and  third,  the  remainders  will  be  as  6  to  7. 

Ans.  30,  48,  50. 

Suggestion.— Let  2a:— 6,  Sx—6,  and  2/ be  the  numbers. 

18.  Two  persons,  A  and  B,  can  perform  a  piece  of  work 
in  16  days.  They  work  together  for  4  days,  when  A,  being 
called  off,  B  is  left  to  finish  it,  which  he  does  in  36  days 
more.     In  what  time  could  each  do  it  separately  ? 

Ans.  A  in  24,  B  in  48  days. 

19.  A  and  B  drink  from  a  cask  of  beer  for  2  hr.,  after 
which  A  falls  asleep,  and  B  drinks  the  remainder  in  2  hr. 
and  48  min. ;  but  if  B  had  fallen  asleep  and  A  had  con- 
tinued to  drink,  it  would  have  taken  him  4  hr.  and  40  min. 
to  finish  the  cask.  In  what  time  could  each  singly  drink 
the  whole?  Ans.  A  in  10,  B  in  6  hrs. 

20.  Di^^ide  the  fraction  |  into  two  parts,  so  that  the 
numerators  of  the  two  parts  taken  together  shall  be  equal 
to  their  denominators  taken  together.         Ans.  i  and  j^. 

21.  A  purse  holds  19  crowns  and  6  guineas.  Now  4 
crowns  and  5  guineas  fill  i|  of  it.  How  many  of  each  will 
it  hold?  Ans.  21  crowns  or  63  guineas. 


108  RAY  S  ALGEBRA,  SECOND  BOOK. 

22.  When  wheat  was  5  shillings  a  bu.  and  rye  3  shil- 
lings, a  man  wanted  to  fill  his  sack  with  a  mixture  of  rye 
and  wheat  for  the  money  he  had  in  his  purse.  If  he 
bought  7  bu.  of  rye  and  laid  out  the  rest  of  his  money  in 
wheat,  he  would  want  2  bu.  to  fill  his  sack ;  but  if  he 
bought  6  bu.  of  wheat,  and  filled  his  sack  with  rye,  he 
would  have  6  shillings  left.  How  must  he  lay  out  his 
money,  and  fill  his  sack  ? 

Ans.  Buy  9  bu.  of  wheat,  and  12  rye. 


SIMPLE    EQUATIONS,    INVOLVING    THREE    OR    MORE    UN- 
KNOWN  QUANTITIES. 

lOO.  Simple  equations,  involving  three  or  more  un- 
known quantities,  may  be  solved  by  either  of  the  three 
methods  of  elimination,  explained  in  Arts.  155  to  159; 
but  the  third  method  is  generally  to  be  preferred. 

1.  Given  5cc— %+2z=r.48,  (1) 
Sx-]-Sy—^z=24,  (2) 
2x — bi/-^Sz^l9,     (3)  to  find  X,  y^  and  z. 

To  eliminate  z  from  (he  first  two  equations,  multiply  (1)  by  2,  and 
then  add  this  to  (2) ,  thus, 

10a:— 8?/+42:=  96,  by  multiplying  (1)  by  2, 
3a;+3j/— 40=  24,     (2) 

13x— 52/        =120,     (5)  by  adding. 

To  eliminate  z  from  equations  (1)  and  (3),  multiply  (1)  by  3,  and 
(3)  by  2,  and  then  subtract;  thus, 

15a;— 12?/+62r=144,  by  multiplying  (1)  by  3, 
4a;— 10?/-|-62r^  38,  by  multiplying  (3)  by  2, 
11a;—  1y        =106,     (6)     by  subtracting. 

To  eliminate  y  from  equations  (5)  and  (6),  multiply  (5)  by  2,  and 
(6)  by  5,  and  then  subtract;  thus, 

55a;— 10r/=530,  by  multiplying  (6)  by  5, 
26a; -10^=240,  by  multiplying  (5)  by  2, 
29a;  =290;  by  subtracting. 

a;=10. 


SIMPLE  EQUATIONS.  109 

110— 2?/— 106,  by  substituting  10  for  x  in  (6);  whence,  2/=2. 
50— 8+22=48,  by  substituting  for  x  and  y  in  (1);  whence,  Z—^. 
It  is  evident  that  the  same  method  may  be  applied  when  the  num- 
ber of  equations  is  four  or  more.     Hence, 

General  Rule  for  Elimination  by  Addition  and  Sub- 
traction— 1st.  Combine  any  one  of  the  equations  with  each 
of  the  others^  so  as  to  eliminate  the  same  unknown  quantity ; 
the  number  of  equations  and  of  unknown  quantities  will  be 
one  less. 

2d.  Combine  any  one  of  these  new  equations  with  each  of 
the  others.,  as  before;  the  7iumber  of  equations  and  of  unknown 
quantities  will  be  two  less. 

3d.  Continue  this  series  of  operations  until  a  single  equation 
is  obtained^  with  one  unknown  quantity.,  and  fnd  its  value. 

4th.  Substitute  this  value  in  the  derived  equations,  for  the 
values  of  the  other  unknoicn  quantities. 

Remark  — In  some  particular  instances,  solutions  maybe  ob- 
tained more  easily  and  elegantly  by  other  means.  As  specimens, 
we  present  the  following: 

2.  Given  — x-^y-\-z=za,  (1) 
x-y^z^b,  (2) 
x-\-y — 2;=c,      (3)   to  find  a:,  y.,  and  z. 

By  adding  the  three  equations  together,  and  calling  a-\^b-\-C=:8, 

We  find  x^y-\-z=8.     (4) 

Then,  by  subtracting  (1),  (2),  and  (3),  respectively  from  (4),  and 
dividing  by  2, 

We  find  .  .  .  x=\{s—a),  =\{b^c). 
y=l{s-b),  =l{a-\-c). 
z=.^{s^c),     =l{a+b). 

In  a  similar  manner,  solve  the  following  examples: 


3.  x+y+z=22, 

y-^z^u^2\, 

iC-|-2-j-«=19, 


Ans.  x=  5, 

U=.    4. 


no 


RAY  S  ALGEBRA,  SECOND  BOOK. 


4.  ^x-y-z=\%  (1)^ 
3i/-x-.=16,  (2)[ 
52— a:— y=24.      (3)  3 

Put  a;4-2/+0=s,  and  add  (1),  (2),  and  (3)  successively  to  the  last 
equation. 
This  gives 3a;=s-fl2    (a) 

4^=5+16     (6) 

60=s+24    (c) 

Multiplying  these  by  4,  3,  and  2,  we  have 

12a;=4s+  48 
12i/^3s+  48 

120=2s+  48 


Or, 


12(a:-f2/+2;)=x9s-f  144,  by  addition. 
.     .     12s^9s-fl44; 
3s==144; 
8=  48. 


Substituting  this  value  of  S  in  (a),  (6),  and  (c),  we  find  iC— 20, 
1^=16,  and  0=12. 


Solve  the  following  by  either  method  of  elimination 
5.  a:-|-3/-f2=6,       "^ Ans.  a:=l, 

4x-f33/-2=Y, 


6.  3x+4^— 5^=32, 

4x— 5^^+32=18, 
^x—'^y—\z=  2. 


7.^_9^+32-10«=21, 
2x+73/-z-«=683, 

3.^+y4_5z-|-2wrrrl95, 

4a;— 6y— 22— 9«=516. 


8.    OJ+^yr^lO— J2, 

i(a;-2):^2y-7. 


y=2, 

2==3. 

Ans.  a;=;10, 
3^=  8, 

z=  6. 

Ans.  crr=100, 

2=:-13, 

.   Ans.  a;^r=7, 
;5:-3. 


SIMPLE   EQUATIONS. 


Ill 


9.  9x—2z-^u^41, 
lz—bu=ll. 


Ans.  xr=.b^ 

3=3, 


Examples  to  be  solved  by  special  methods  : 


10. 


1^1 


X 

X       z 

y    ^ 


=h. 


Ans.  X- 


_2_ 
a-f-6— c' 

2 

2 

b-{-c — a 


Suggestion.— Subtract  (3)  from  (2),  then  combine  the  resulting 
equation  with  (1),  to  find  X  and  y;  z  may  be  found  similarly. 


11.    ~x-Yij^z^v=a, 
x—y-\rZ-\-v=^h, 

x-\-y Z-\-V:=C^ 

x-\-y-\-z—v=d, 


Ans.  a:=J(s — a), 
y=l{s-h), 


wlierc  s=:\(^a-\-h-{-c-\-d'). 


PROBLEMS  PRODUCING   SIMPLE    EQUATIONS    CONTAINING 
THREE.  OR  MORE  UNKNOWN  QUANTITIES. 

1G1«  For  the  method  of  forming  the  equations,  see  Arts.  154 
and  159. 

1.  The  stock  of  three  traders  amounts  to  $760  ;  the 
shares  of  the  first  and  second  exceed  that  of  the  third 
by  $240  ;  and  the  sum  of  the  second  and  third  exceeds  the 
first  by  $360;  what  is  the  share  of  each  ? 

Ans.  $200,  $300,  and  $260. 

2.  What  three  numbers  are  there,  each  greater  than  the 
preceding,  whose  sum  is  20,  and  such  that  the  sum  of  the 
first  and  second  is  to  the  sum  of  the  second   and   third, 


112  RAY'S  ALGEBRA,  SECOND  BOOK. 

as  4  is  to  5  ;  and  the  difference  of  the  first  and  second,  is 
to  the  difference  of  the  first  and  third,  as  2  to  3? 

Ans.  5,  7,  and  8. 

3.  Find  four  numbers,  such  that  the  sum  of  the  first, 
second,  and  third  shall  be  13;  the  sum  of  the  first,  second, 
and  fourth,  15  ;  the  sum  of  the  first,  third,  and  fourth,  18  ; 
and  lastly,  the  sum  of  the  second,  third,  and  fourth,  20. 

Ans.  2,  4,  7,  9. 

4.  The  sum  of  three  digits  composing  a  certain  number 
is  16  ;  the  sum  of  the  left  and  middle  digits,  is  to  the  sum 
of  the  middle  and  right  ones  as  3  to  3g  ;  and  if  198  be 
added  to  the  number,  the  order  of  the  digits  will  be  in- 
verted.    Required  the  number.  Ans.  547. 

5.  It  is  required  to  find  tliree  numbers  such  that  i  the 
first,  ^  the  second,  and  j  the  third,  shall  together  make 
46  ;  J  the  first,  |  the  second,  and  4  the  third,  shall  to- 
gether make  35  ;  and  |  the  first,  i  the  second,  and  i  the 
third,  shall  together  make  28 1.       Ans.  12,  60,  and  80. 

6.  The  sum  of  three  numbers,  taken  two  and  two,  are 
a,  ?>,  and  c.     What  are  the  numbers  ? 

Ans.  -i(a+6— c),  K^+c—h),  and  ^(h-^c—a). 

7.  A  person  has  four  casks,  the  second  of  which  being 
filled  from  the  first,  leaves  the  first  ^  full.  The  third  being 
filled  from  the  second,  leaves  it  J  full ;  and  when  the  third 
is  emptied  into  the  fourth,  it  is  found  to  fill  only  -f^  of  it. 
But  the  first  will  fill  the  third  and  fourth  and  have  fifteen 
quarts  remaining.     How  many  quarts  does  each  hold  ? 

Ans.  140,  60,  45,  and  80,  respectively. 

8.  In  the  crew  of  a  ship  consisting  of  sailors  and  sol- 
diers, there  were  22  sailors  to  every  3  guns,  and  10  sailors 
over ;  also  the  whole  number  of  hands  was  5  times  the 
number  of  soldiers  and  guns  together  ;  but  after  an  engage- 
ment, in  which  the  slain  were  one  fourth  of  the  survivors, 
there  wanted  5  men  to  be  13  men  to  every  2  guns.  Re- 
quired the  number  of  guns,  soldiers,  and  sailors. 

Ans.  90  guns,  55  soldiers,  670  sailors. 


SIMPLE  EQUATIONS.  113 


V.    SUPPLEMENT     TO     SIMPLE 
EQUATIONS. 

Remark. — The  principles  employed  in  algebraic  equations  ma; 
be  variously  applied.  We  may,  for  example,  by  their  aid  demon- 
strate several  of  the  theorems   in  fractions.     Thus,  to   prove  that 

—r  =  -r,     Art.  118);    put  q=j. 

♦ 

Then,  bq=a,  ^ndmbq=ma]  ..  q=r — 

a        ,        ma       ma     a 

Hence,  since  0'— 7-,  and  g=-    ,,  .-. — j- = -?-• 

'  ^6  mo       mb       b 

To  prove  that  ^X^=  5^,   (Art.  131);  put  p=-^  and  g-^. 

Then,  bp—a,  and  dq^C.     Multiplying  the  last  two  equations, 

member  by  member, 

,  ^  <^C       ,  .  ,  ,         , 

We  have  bdpq—ac\  .-.  pq=j-j,  which  proves  the  rule. 

In  a  similar  manner,  the  rules  for  Addition,  Subtraction,  and 
Division  of  fractions  may  be  demonstrated. 

Other  methods  of  application  are  given  in  Arts,  following. 


I.    GENERALIZATION. 

16!S.  Literal  Equations  are  those  in  -which  the  known 
quantities  are  represented,  either  entirely  or  partly,  by 
letters. 

Quantities  represented  by  letters  are  termed  general 
values,  because  the  solution  of  one  problem  furnishes  a 
general  solution. 

A  Formula  is  the  answer  to  a  problem,  when  the  known 
quantities  are  represented  by  letters. 

A  Rule  is  a  formula  expressed  in  ordinary  language. 
2d  Bk.  10 


114  RAYS  ALGEBRA,  SECOND  BOOK, 

By  the  application  of  Algebra  to  the  solution  of  general 
questions,  many  useful  and  interesting  truths  and  rules  may 
be  established.     Take  the  following  as  an  example : 

163.  Divide  a  given  number  a  into  three  parts,  having 
to  each  other  the  same  ratio  as  the  numbers  m,  «,  and  p. 

Let  mx,  nx,  and  px^  represent  the  required  parts. 
Then, mx^nx-^-px^a, 

a 
And  ....     iC=: ;  from  which  we  obtain, 

ma  na  .  pa 

mx= ,   nx= ,  and  px— — — 

m-\-n^p  m-i^ri-^-p  ^      'm-\-n-]-p 

This  formula,  expressed  in  words,  gives  the  following 
Rule  for  Dividing  a  Given  Number  into  Parts  having 

to  each  other  a  Given  Ratio. — Multiply  the  given  nnmher 
hy  each  term  of  the  ratios  respectively^  and  divide  the  prod- 
nets  hy  the  sum  of  the  numbers  expressing  the  ratios. 

Solve  the  following  examples  by  this  rule,  and  test  its 
accuracy  by  verifying  the  results : 

2.  Divide  69  into  three  parts,  having  to  each  other  the 
game  ratio  as  the  numbers  5,  7,  and  11. 

Ans.  15,  21,  and  33. 

3.  Divide  38]  into  four  parts,  having  to  each  other  the 
same  ratio  as  the  fractional  numbers  ],  i,  ],  and  1. 

Ans.  15,  10,  1l,  and  6. 

Solve  the  following  general  examples,  express  the  formula  in 
ordinary  language,  so  as  to  form  a  general  rule,  and  apply  the  rule 
or  the  formula,  to  the  solution  of  the  numerical  problems. 

4.  The  sum  of  two  numbers  is  a,  and  their  difference  h. 

Required  the  numbers.        .     _,  ah.         ah 

A.  Greater,  ^-hg  ;  less,  ^  —  ^. 


SIMPLE  EQUATIONS.  115 

5.  The  joint  capital  of  A  and  B  in  a  firm,  is  $16000; 
but  A's  investment  is  $2000  more  than  B's.  Required  the 
capital  of  each.  Ans.  A's,  $9000  ;  B's,  $7000. 

6.  The  sum,  of  two  angles  is  120°  44'  52'',  and  their 
difi"erence  is  26°  32'  18".     Required  the  angles. 

Ans.  Greater,  73°  38'  35";  less,  47°  6'  17". 

7.  The  difference  of  two  numbers  is  a,  and  the  greater 
is  to  the  less  as  m  to  n.     Find  the  numbers, 

Ans.  - 


8.  The  difference  in  capacity  of  two  cisterns  is  678  gal., 
and  the  greater  is  to  the  less  as  7  to  5.  How  much  does 
each  hold?  Ans.  Greater,  2373  gal.  ;  less,  1695. 

9.  The  sum  of  two  numbers  is  a,  and  their  sum  is  to 
their  difference  as  m  to  n.     Required  the  numbers. 

Ans.  Greater,  :=:-    ~T        ;  less,  ^^-^ — -  . 

10.  An  estate,  valued  at  $8745,  was  divided  between  a 
son  and  daughter  in  such  a  manner  that  the  sum  of  their 
shares  was  to  the  difference  as  5  to  3.  What  was  the  share 
of  each?  Ans.  Son's,  $6996  ;  daughter's,  $1749. 

11.  Divide  the  number  a  into  three  such  parts,  that  the 

second  shall  exceed  the  first  by  6,  and  the  third  exceed  the 

second  by  c.  a — 2h — c     a-\-h — c     a-\-h-\-2c 

Ans.  g        ,  g— ,  g        . 

12.  At  a  certain  election  the  whole  number  of  votes  cast 
was  602.  B  received  84  more  votes  than  A,  and  C  56  more 
than  B.     How  many  did  each  receive  ? 

Ans.  A  126,  B  210,  C  266. 

13.  Divide  the  number  a  into  four  such  parts,  that  the 
first  increased  by  m,  the  second  diminished  by  m,  the  third 
multiplied  by  w,  and  the  fourth  divided  by  m,  shall  be  all 
equal  to  each  other. 

.  ma  ma        .  a  m^a 


(wi+l)2        !  (m-{-iy^    '  (m-j-1)^'    (m+l)2' 


liG  RAYS  ALGElillA,  SECOND  DOOK. 

Let  the  four  parts  be  represented  by  X — m,  x-\-m^  — ,  and  mx. 

14.  Divide  the  number  245  into  four  parts,  such  that 
the  first  increased  by  6,  the  second  diminished  by  6,  the 
third  multiplied  by  6,  and  the  fourth  divided  by  6,  shall 
be  all  equal  to  each  other.         Ans.  24,  36,  5,  and  180. 

15.  A  person  has  just  a  hours  at  his  disposal;  how  far 

may  he  ride  in  a  coach  which  travels  h  miles  an  hour,  so  as 

to  return  home  in  time,  walking  back  at  the  rate  of  c  miles 

an  hour?  .  ahc       .. 

Ans.  ,   ,      miles. 
6-f  c 

16.  A  person  finds  that  he  can  row  a  skiflf  6  miles  an 
hour  with  the  current,  and  3  miles  an  hour  against  it ;  how 
far  can  he  pass  down  the  stream,  and  yet  return  to  the  point 
from  which  he  set  out,  in  8  hours?  Ans.  16  miles. 

17.  Given  the  sum  of  two  numbers  =«,  and  the  quotient 
of  the  greater  divided  by  the  less  =6.  Required  the  num- 
bers. .         ^  a  ah 


This  gives  the  following  simple  rule:  To  find  the  less  nurnber,  divide 
the  sum  of  the  numbers  by  (heir  quotient  increased  by  unity. 

18.  The  sum  of  two  numbers  is  256,  and  the  quotient 
of  the  greater  by  the  less  is  15.     Required  the  numbers. 

Ans.  240  and  16. 

19.  A  person  distributed  a  cents  among  n  beggars,  giv- 
ing h  cents  to  some,  and  c  to  the  rest.     How  many  were 

there  of  each  ?      .        a — nc       ,  ,  nh — a 

Ans.  -,- at  h  cts.,  and  —, at  c  cts. 

6 — c  b — c 

20.  A  father  divided  $8500  ^mong  7  children,  giving 
to  each  son  $1750,  and  to  each  daughter  $500.  How 
many  of  his  children  were  sons  and  how  many  daughters  ? 

Ans.  4  sons,  8  daughters. 


SIMPLE  EQUATIONS.  117 

21.  Divide  the  number  n  into  two  such  parts,  that  the 

quotient  of  the  greater  divided  by  the  less  shall  be  q,  with 

a  remainder  r.  nq-\~r     n — ?' 

Ans.  ^^  ^i . 

1+2'    1+? 

22.  Divide  1903  into  two  such  parts  that  the  quotient 
of  the  greater  divided  by  the  less  shall  be  12,  with  a  re- 
mainder 5.  Ans.  1757  and  146. 

23.  If  A  and  B  together  can  perform  a  piece  of  work 

in  a  days,  A  and  C  in  ^  days,  and  B  and  C  in  c  days :  find 

the  time  in  which  each  can  perform  it  separately. 

.  .   .  2ahc  _,  .  2ahc  „  .  2ahc 

Ans.  A  in  — — ,  B  in  -j-^ ,  C  in  -— da. 

ac-\-bc — ab  ab-\-uc — ac  ab-\-ac — be 

24.  A  tank  is  supplied  with  water  from  three  pumps. 
The  first  and  second  will  fill  it  in  30  hours,  the  first  and 
third  in  40  hours,  and  the  second  and  third  in  50  hours. 
in  what  time  can  each  fill  it  separately  ? 

Ans.  First  in  52rf*3,  second  in  70j?,  third  in  l7l|  hrs. 

25.  A,  B,  and  C  hold  a  pasture  in  common,  for  which 
they  pay  P  $  a  year.  A  puts  in  a  oxen  for  m  months ;  B, 
h  oxen  for  n  months  ;  and  C,  c  oxen  for  p  months.  Re- 
quired each  one's  share  of  the  rent. 

Ans.  A's, ^-^^—  P  $ ;  B's, -^- —  P%;  and 

CX  -^--  P%. 

ma-\-nb-{-pc 

From  these  formulas  is  derived  the  rule  of  Compound  Fellowship. 


26.  A,  B,  and  C  engage  in  business  together.  A  put 
into  the  firm  $600  for  30  weeks,  B  $500  for  40  weeks, 
and  C  $800  for  28  weeks.  They  then  divided  a  profit  of 
$1812  between  them.     What  was  each  man's  share  ? 

Ans.  A's,  $540;  B's,  $600;  C's,  $672. 

27.  A  mixture  is  made  of  a  lb.  of  tea  at  m  shillings  per 


118  RAY'S  ALGEBRA,  SECOND  BOOK. 

lb.,  b  lb.  at  n  shillings,  and  c  lb.  at  p  shillings  :  what  will 

be  its  cost  per  lb.  ?  ^         ??m-f  w/>4-«c 

Ans.  ,  .   ,        . 

From  this  formula  is  derived  the  rule  termed  Alligation  Medial. 

28.  A  drover  bought  10  cattle  at  $30  apiece,  12  at 
$40,  and  8  at  $90.     What  was  the  average  price  per  head  ? 

Ans.  $50. 

29.  A  waterman  rows  a  given  distance  a  and  back  again 
in  h  hours,  and  finds  that  he  can  row  c  miles  with  the 
current  for  d  miles  against  it :  required  the  times  of  rowing 
down  and  up  the  stream,  also  the  rate  of  the  current  and 
the  rate  of  rowing. 

Ans.  Time  down,  — j — - ;  time  up,  — -—  ;  rate  of  current, 

— ^^vi — r-^:  rate  of  rowinp',     \,    /  . 
2hcd     '  ^'      2bcd 

30.  A  vessel  sailed  with  the  wind  and  tide  60  miles,  and 
returned  ivith  the  wind  and  against  the  tide.  She  reached 
the  same  point  in  12  hours,  and  the  rate  of  sailing  out  and 
in  was  as  5  to  3.  Required  the  time  each  way,  and  the 
strength  of  the  wind  and  tide. 

Ans.  Time  out,  4\  hours ;  time  in,  7^  hours ;  wind, 
10^  miles  per  hour ;  tide,  2|  miles  per  hour. 


II.    NEGATIVE    SOLUTIONS. 

164.  It  sometimes  happens,  in  the  solution  of  a  prob- 
lem, that  the  result  has  the  minus  sign.  This  is  termed  a 
negative  solution.  We  shall  now  examine  a  question  of 
this  kind. 

1.  What  number  must  be  subtracted  from  3  that  the  re- 
mainder may  be  7  ? 

Let x=  the  number. 

Then,  3—X—  7;  whence,  —x=7—3;  or,  z  =—4. 


SIMPLE  EQUATIONS.  119 

Now,  — 4  subtracted  from  3,  gives  a  remainder?;  and  the  an- 
swer, — 4,  is  said  to  satisfy  the  question  in  an  algebraic  sense. 

The  problem  is  evidently  impossible  in  an  arithmetical  sense,  and 
this  impossibility  is  shown  by  the  negative  answer.  But,  since  sub- 
tracting —4  is  the  same  as  adding  -|-4  (Art.  48),  the  result  is  the 
answer  to  the  following: 

What  number  must  be  added  to  3,  that  the  sum  may  be  equal  to  7  ? 

Let  the  question  now  be  generalized,  thus : 

What  number  must  be  subtracted  from  a,  that  the  re- 
aaainder  may  be  6? 

Let X:=  the  number. 

Then,  a—Xz=b\  whence,  x=a—b. 

While  b  is  less  than  a,  the  value  of  x  will  be  positive;  and  the 
question  will  be  consistent  in  an  arithmetical  sense. 

But  if  b  becomes  greater  than  a,  the  value  of  x  will  be  negative; 
and  the  question  will  be  consistent  in  its  algebraic,  but  not  in  its 
arithmetical  sense. 

When  b  becomes  greater  than  a,  the  question,  to  be  consistent  in 
an  arithmetical  sense,  should  read : 

What  number  must  be  added  to  a  that  the  sum  may  be 
equal  to  6? 

From  this  we  derive  the  following  important  general  principles r 

1st.  A  negative  solidwn  indicates  smne  arithmetical  incorv- 
sistency  or  absurdity  in  tJw  qv£stionfrom  which  the  equation  ima 
derived. 

2d.  When  a  negative  solution  is  obtained,  Hie  question,  to 
which  it  is  the  answer,  may  be  so  modified  as  to  be  consistent 
with  arithmetixnl  notions. 

After  solving  the  following  questions,  let  them  be  so  modified  that 
the  results  may  be  true  in  an  arithmetical  sense. 

2.  What  number  must  be  added  to  the  number  30,  that 
the  sum  may  be  19?  (x= — 11). 

3.  The  sum  of  two  numbers  is  9,  and  their  difference  25; 
required  the  numbers.  Ans.  1*7  and  — 8. 


120  RAYS  ALGEBRA,  SECOND  BOOK. 

4.  What  number  is  that  whose  half  subtracted  from  its 
third  leaves  a  remainder  15?  (x= — 90). 

5.  A  father's  age  is  40  years;  his  son's  age  is  13  years; 
in  how  many  years  will  the  age  of  the  father  be  4  times 
that  of  the  son?  (cc=: — 4). 

III.    DISCUSSION    OF    PROBLEMS. 

lOS.  After  a  question  has  been  generalized  and  solved, 
we  may  inquire  what  values  the  results  will  have,  when 
particular  suppositions  are  made  with  regard  to  the  known 
quantities. 

The  determination  of  these  values,  and  the  examination 
of  the  various  results,  to  which  different  suppositions  give 
rise,  constitute  the  discussion  of  the  problem. 

The  various  forms  which  the  value  of  the  unknown 
quantity  may  assume,  are  shown  in  the  discussion  of  the 
following : 

1.  After  subtracting  h  from  a,  what  number,  multiplied 
by  the  remainder,  will  give  a  product  equal  to  c  ? 

Let  x=  the  number. 

Then,  {a—b)x=c^  and  x= 


a—b' 


This  result  may  have  five  different  forms,  depending  on 
the  different  values  that  may  be  given  to  «,  fe,  and  c. 

To  express  these  forms,  let  A  denote,  indefinitely,  some 
quantity. 

I.  When  h  is  less  than  a.  In  this  case,  since  a — h  will 
be  positive,  the  value  of  x  will  be  of  the  form  -|-A. 

II.  When  h  is  greater  than  a.  In  this  case,  a — h  will 
be  negative,  and  the  value  of  ic,  of  the  form  — A. 

III.  When  h  is  equal  to  a.  In  this  case,  tho  value  of 
X  is  of  the  form  -^,  or,  (Art.  136),  x=^sxi , 


SIMPLE  EQUATIONS.  121 

IV.  When  c  is  0,  and  b  either  greater  or  less  than  a. 

In  this  case,  the  value  of  x  is  of  the  form  — ,  or,  (Art, 

A  ^ 

136),  x=0. 

V.  When  b  is  equal  to  a,  and  c  is  equal  to  0.  In  this 
case,  the  value  of  x  is  of  the  form  ^,  which  (Art.  13^)  is 
the  symbol  of  indetermination. 

The  discussion  of  the  following  problem,  originally 
proposed  by  Clairaut,  will  serve  to  illustrate  further  the 
preceding  principles,  and  show  that  the  results  of  every 
correct  solution  correspond  to  the  circumstances  of  the 
problem. 

PROBLEM    OF    THE    COURIERS. 

1G6.  Two  couriers  depart  at  the  same  time,  from  two 
places,  A  and  B,  distant  a  miles  from  each  other ;  the 
former  travels  m  miles  an  hour,  and  the  latter  n  miles : 
where  will  they  meet? 

There  are  two  cases  of  this  problem,  according  as  the 
couriers  travel  toward  each  other,  or  in  the  same  direc- 
tion. 

I.  When  the  couriers  travel  toward  each  other. 

Let  P  be  the  point  where  they  meet,  A  Im^immmm^mmmhkI  B 
and  a=AB,  the  distance  between  the  P 

two  places. 

Let  a;=AP,  the  distance  which  the  first  travels. 

Then,  a — iC=BP,  the  distance  which  the  second  travels. 

But  the  distance  each  travels,  divided  by  the  number  of  miles  trav- 
eled per  hour,  will  give  the  number  of  hours  he  was  traveling. 

X 
Therefore,    — =  the  number  of  hours  the  first  travels. 
m 

And =  "         "       "        "     second  travels. 

n 

2d  Bk.  11* 


122  RAY'S  ALGEBRA,  SECOND  BOOK. 

But  they  both  travel  the  same  number  of  hours ;  therefore, 

X      a-x 

m~'    n    ' 

nx^7na—7nx\ 

ma  ,  na 

and  a — Xz 


1st.    Suppose   m—ri]  then,  0:=^^  ^^o'  and  a— x=-^;  that  is, 
if  they  travel  at  the  same  rate,  each  travels  half  the  distance. 

2d,  Suppose  71=0;  then,  x=z  —  =^a\  that  is,  if  the  second  cou- 
rier remains  at  rest,  the  first  travels  the  whole  distance  from  A  to  B. 
In  like  manner,  if  m=0,  a—X—a. 

3d.  Suppose  rn^n,  then  the  value  of  x  will  be  greater  than  that 
of  a— a:,  since  ma  is  greater  than  na;  that  is,  the  point  P  will  be 
farther  from  A  than  B  if  m<^n,  then  the  value  of  x  will  be  less 
than  that  of  a — X,  or  P  will  be  nearer  A. 

All  of  these  results  are  evidently  true,  and  correspond  to  the  cir- 
cumstances of  the  problem. 

II.  When  the  couriers  travel  in  the  same  direction. 
As  in  the  first  case,  let  P  be  the  point 


of  meeting,  each  traveling  from  A  toward        -^  ■" 

P,  and  let  a=AB,  the  distance  between  the  places; 
a:=-AP,  the  dhtance  the  first  travels ; 
x—a=BF,   the  distance  the  second  travels. 

Then,  reasoning  as  in  the  first  case,  we  have 

X  _^— «, 

m"    n    ' 

nx  =  mx — ma ; 

m^a  ,  na 

X  = ;    and  x—a-. 


m—n'  m—n 

1st.  If  we  suppose  m  greater  than  w,  the  values  of  X  and  of  x—a 
will  both  be  positive ;  that  is,  the  couriers  will  meet  on  the  right 
of  both  A  and  B.  This  evidently  corresponds  to  the  circumstances 
of  the  problem. 

2d.  If  we  suppose  n  greater  than  w,  the  value  of  x,  and  also  that 
of  x—a,  will  be  negative. 


SIMPLE  EQUATIONS.  123 

Now,  since  the  positive  values  of  x  and  X — a  implied  that  the 
couriers  met  at  a  point  P,  on  the  riyht  of  A  and  B,  the  negative 
values  must  indicate  (Art.  47)  that  the  place  of  meeting  is  at  P^,  on 

I  1  1  i 

V  A  B  p 

the  left  of  A  and  B.  Indeed,  where  m  is  less  than  n,  or  the  advance 
courier  is  traveling  faster  than  the  other,  it  is  evident  that  they  can 
not  meet  in  the  future.  We  may,  however,  suppose  that  they  have 
met  before.  We  may,  therefore,  on  the  principles  explained  in 
Art.  164,  modify  the  question  in  one  of  two  ways. 

1st.  We  may  inquire,  Where  have  the  couriers  met?  or, 
2d.  We  may  suppose  the  direction  changed^  and  call  A  the  advance 
courier;  that  is,  that  they  travel  toward  P''.     We  shall  then  have 
AB=:a,  AP^=a:,  and  BP^— a-f  a;.     Forming  and  solving  the  equa- 
tion as  before,  we  should  obtain  positive  values  of  x  and  a-\-x. 
3d.  If  we  suppose  m  equal  to  n;  then, 

ma  ,  na 

x=-^=  cc,  and  a:— a=— =  oo. 

This  evidently  corresponds  to  the  circumstances  of  the  problem; 
for,  if  the  couriers  travel  at  the  same  rate,  the  one  can  never  over- 
take the  other.  This  is  sometimes  expressed,  by  saying  they  only 
meet  at  an  infinite  distance  from  the  point  of  starting. 

4th.  If  we  suppose  a=0;  then, 

x= =0,  and  x—a= =0. 

m—n  m — n 

This  corresponds  to  the  circumstances  of  the  problem ;  for,  if  the 
couriers  are  no  distance  apart,  they  will  have  to  travel  no  (0)  dis- 
tance to  be  together. 

5th.  If  we  suppose  m=^n^  and  a=:^0;  then,  a;=j{,  and  x—a=^. 

But  this  is  the  symbol  of  indeterminatxon^  and  indicates  (Art.  137) 
that  the  unknown  quantity  may  have  any  finite  value  whatever. 
This,  also,  evidently  corresponds  to  the  circumstances  of  the  prob- 
lem; for,  if  the  couriers  are  no  distance  apart,  and  travel  at  the 
same  rate,  they  will  be  always  together ;  that  is,  at  any  distance  what- 
ever from  the  point  of  starting. 

6th.  If  we  suppose  w=0;  then,  x= —  =a;  that  is,  the  first  cou- 
rier travels  from  A  to  B,  overtaking  the  second  at  B.  So,  if  m==Q, 
x—a^=—a. 


124  RAY'S  ALGEBRA,  SECOND  BOOK. 

7th.  If  we  suppose  their  rate  of  travel  has  a  given  ratio,  as 
n=-^ ;  then,  X= =^za;  that  is,  the  first  travels  twice  the  dis- 
tance from  A  to  B  before  overtaking  the  second.  The  results  in 
the  last  two  cases  evidently  correspond  to  the  circumstances  of  the 
problem. 


IV.   CASES  OF  INDETERMINATION  IN   SIMPLE   EQUATIONS 
AND  IMPOSSIBLE  PROBLEMS. 

167.  An  Independent  Equation  is  one  in  which  the 
relation  of  the  quantities  which  it  contains  can  not  be  ob- 
tained directly  from  others  with  which  it  is  compared. 

Thus, a;4- 3^=19, 

are  equations  which  are  independent  of  each  other,  since  the  one 
can  not  be  obtained  from  the  other  in  a  direct  manner. 
a:+3i/^19, 
2a:-f62/=:38, 
are  not  independent  of  each  other,  the  second  being  derived  directly 
from  the  first,  by  multiplying  both  sides  by  2. 

168.  An  Indeterminate  Equation  is  one  that  can  be 
verified  by  different  values  of  the  same  unknown  quantity. 

Thus,  if  we  have,     .     .     .     X— 2/=3, 

By  transposition,     .     .     .     Xz=z^^-\-y. 

If  we  make  y=\\  then,  a;=4.  If  we  make  2/=^2;  then,  rc=5, 
and  so  on;  from  which  it  is  evident  that  an  unlimited  number  of 
Talues  may  be  given  to  x  and  y^  that  will  verify  the  equation. 

If  we  have  two  equations  containing  three  unknown 
quantities,  we  may  eliminate  one  of  them  ;  this  will  leave 
one  equation  containing  two  unknown  quantities,  which,  as 
in  the  preceding  example,  will  be  indeterminate. 

Thus,  in rr+Sy— 5^=20, 

X-  2/+30=16, 
If  we  eliminate  a:,  we  have,  after  reducing, 

y—2z=l ;  whence,  y=l-\-2z. 


SIMPLE  EQUATIONS.  125 

If  we  make  0=1  ;  then,  2/=3,  and  a:=20-f  52r— 3?/=16.  If  we 
make  s=2;  then,  ^=5,  and  x=:15. 

So,  any  number  of  values  of  the  three  unknown  quantities  may- 
be found  that  will  verify  both  equations.  These  examples  are  suf- 
ficient to  establish  the  following 

General  Principle. —  When  the  number  of  unknown  quan- 
titles  exceeds  the  number  of  independent  equations^  the  prob- 
lem is  indeterminate. 

A  question  that  involves  only  one  unknown  quantity  is 
sometimes  indeterminate  ;  the  equation  deduced  from  the 
conditions  being  identical.  (Art.  145.)  The  following  is 
an  example : 

What  number  is  that,  whose  \  increased  by  the  J  is 
equal  to  the  i^  diminished  by  the  f-^'i 

Let  X:=  the  number. 
_,        XX     11a:     2x 
Then,  ^  +  6=20r-l5' 

Clearing  of  fractions,  15a:-f  10a:=33a;— 8a;;  or,  2oa;=25a;;  which 
will  be  verified  by  any  value  whatever  of  x. 

169.  The  reverse  of  the  preceding  case  requires  to  be 
considered ;  that  is,  when  the  number  of  equations  is 
greater  than  the  number  of  unknown  quantities. 

Thus,  we  may  have  2a:-f  3i/=23  (1.) 
^x-2y=  2  (2.) 
5a:+42/=40        (3.) 

Each  of  these  equations  being  independent  of  the  other 
two,  one  of  them  is  unnecessary,  since  the  values  of  x 
and  ;y,  may  be  found  from  either  two  of  them. 

When  a  problem  contains  more  conditions  than  are 
necessary  for  determining  the  values  of  the  unknown 
quantities,  those  that  are  unnecessary  are  termed  re- 
dundant. 

The   number   of  equations  may  exceed  the  number  of 


126  RAY'S  ALGEBRA,  SECOND  BOOK. 

unknown  quantities,  so   that  the  values   of  the  unknown 
quantities  shall  be  incompatible  with  each  other. 

Thus,  if  we  have  x-\-  y=l2,  (1.) 
2a:+  y^ll  (2.) 
3a:-f22/=30        (3.) 

The  values  of  X  and  ?/,  found  from  equations  (1)  and  (2),  are 
a:=5,  y^^^l-^  from  (1)  and  (3),  a;^G,  and  2/=6;  and  from  (2)  and 
(3),  a:i=4,  and  y='^.  It  is  manifest  that  only  two  of  these  equa- 
tions can  be  true  at  the  same  time. 


EXAMPLES    TO    ILLUSTRATE    THE    PRECEDING 
PRINCIPLES. 

1.  What  number  is  that,  which  being  divided  succes- 
sively by  m  and  n^  and  the  first  quotient  subtracted  from 

the  second,  the  remainder  shall  be  o  ?  .  mnq 

Ans.  x=. -. 

m — n 

What  supposition  will  give  a  negative  solution  ?  Will  any  sup- 
position give  an  infinite  solution?  An  indeterminate  solution? 
Illustrate  by  numbers. 

2.  Two  boats,  A  and  B,  set  out  at  the  same  time,  one 

from   C   to   L,  and   the   other  from   L   to   C  ;   the  boat  A 

runs  m  miles,  and  the  boat  B,  n  miles  per  hour.     Where 

will  they  meet,  supposing  it  to  be  a  miles  from  C  to  L  ? 

ma        .  na      .    „         ^ 

Ans.  — — -   mi.  irom  C,  or mi.  from  L. 

m-\-n  m-\-n 

Under  what  circumstances  will  the  boats  meet  half  way  between 
C  and  L?  Under  what  will  they  meet  at  C  ?  At  L?  Above  C? 
Below  L?  Under  what  circumstances  will  they  never  meet? 
Under  what  will  they  sail  together  ?     Illustrate  by  numbers. 

3.  Given  2x—y=2,  5x— 3^=3,  ^x^2y=Vl,  4rr4-3y 
=24 ;  to  find  x  and  y,  and  show  how  many  equations  are 
redundant.     (Art.  169.)  Ans.  a;=3,  y=4. 

4.  Given  a;+2y=ll,  2.t— 7/=Y,  3a;— ^^=17,  a;+3^=19; 
to  show  that  the  equations  are  incompatible.     (Art.  169  ) 


SIMPLE  EQUATIONS.  127 

V.    A    SIMPLE    EQUATION    HAS    BUT    ONE    ROOT. 

ITO.  Any  simple  equation  involving  only  one  unknown 
quantity,  (x),  may  be  reduced  to  the  form  mx=n;  for  all 
the  terms  containing  x  may  be  reduced  to  one  term,  and 

11 
all  the  known  quantities  to  one  term:  whence,  a;=-. 
^  m 

Now,  since  n  divided  by  m  can  give  but  one  quotient, 
we  infer  that  a  simple  equation  has  hut  one  root;  that  is, 
there  is  but  one  value  that  will  verify  the  equation. 


VI.    EXAMPLES    INVOLVING    THE   SECOND  POWER 
OF    THE    UNKNOWN    QUANTITY. 

ITl.  It  sometimes  happens  in  the  solution  of  an  equa- 
tion, that  the  second  or  some  higher  power  of  the  unknown 
quantity  occurs,  but  in  such  a  manner  that  it  is  easily 
removed. 

The  following  equations  and  problems  belong  to  this 
class : 

1.  {^^x)(x—b)=(x—2y. 

Performing  the  operations  indicated,  we  have 
a;2_a;_20=a:-— 4a;^4. 

Omitting  X^  on  each  side,  and  transposing,  we  have 
3a:^24,  or  xS. 

'  ^fe?+4=^+i ^-  -'■ 


6x— 43' 


4.    .r  Vo(3:r-19)^2a;4-19 Ans.  a;. 


-     aW^x:')  ,   ax  .  ^ 

5.     -^- ^=ac+  -^ Ans.  x=^-. 

hx  ^  h  G 


128  RAY'S  ALGEBRA,  SECOND  BOOK 

^        ex"*         dx"^  .  ad — ce 

O.     —-7-=  ~r-j- Ans.  x==  ,.     ■  .. 

a-\-ox     €-\-/x  cf — ud 

7.  It  is  required  to  find  a  number  whicli  being  divided 
into  2  and  into  3  equal  parts,  4  times  the  product  of  the 
2  equal  parts  shall  be  equal  to  the  continued  product  of 
the  3  equal  parts.  Ans.  27. 

8.  A  rectangular  floor  is  of  a  certain  size.  If  it  were 
5  feet  broader  and  4  feet  longer,  it  would  contain  116  feet 
more  ;  but  if  it  were  4  feet  broader  and  5  feet  longer,  it 
would  contain  113  feet  more.  Required  its  length  and 
breadth.  Ans.  Length,  12  feet  j  breadth,  9  feet. 


YI.    OF    POWERS,   ROOTS,   RADICALS, 
AND    IJ^EQUALITIES. 

I.    INVOLUTION,    OR    FORMATION    OF    POWERS. 

1T!S.  The  Power  of  a  number  is  the  product  obtained 
by  multiplying  it  a  certain  number  of  times  by  itself. 

Any  number  is  the  first  power  of  itself. 

When  the  number  is  taken  twice  as  a  factor,  the  product 
is  called  the  second  power  or  square  of  the  number. 

When  the  number  is  taken  three  times  as  a  factor,  the 
product  is  called  the  third  power  or  cube  of  the  number. 

In  like  manner,  the  fourth^  fifih,  etc.,  powers  of  a  num- 
ber are  the  products  arising  from  taking  the  number,  as  a 
factor,  four  times,  five  times,  etc. 

The  Index  or  Exponent  of  the  power  is  the  number 
which  denotes  the  power.  It  is  written  to  the  right  of 
the  number,  and  a  little  above  it. 


FORMATION  OF  POWERS.  129 


Thus, 


3=31- 

.    3,  is 

the 

let  power 

of  3. 

3X3=32= 

=     9,  - 

2d 

u 

"  3. 

3X3X3=33= 

=  27,  » 

3d 

u 

»  3. 

3X3X3X3=3^= 

.  81,  " 

4th 

u 

"  3. 

ixfxfx|=(|)^= 

=  &'" 

4th 

u 

"|. 

aX«X«X«=(«)^- 

-a*      " 

4th 

« 

"  a. 

ac 

:2xac2xac2^(«c-)'''= 

=a-V^  " 

3d 

u 

ac^. 

Prom  the  above,  we  have  the  following 

General  Rule  for  Raising  any  Quantity  to  any  Required 

Power. — Multiply  the  given  quantity  hy  itsclj]  until  it  is  taken 
as  a  factor  as  many  times  as  there  are  units  in  the  exponent 
of  the  required  power. 

As  the  application  of  this  general  rule  frequently  involves 
a  tedious  operation,  it  is  best  to  reduce  the  labor  attending 
it.  It  will,  therefore,  be  most  convenient  to  divide  the 
subject  into  distinct  cases. 

Case  I. — To  raise  a  Monomial  Quantity  to  any 
Power. 

By  inspecting  the  illustration  above  given,  it  will  be 
seen  that  a  coefficient  is  involved  by  repeated  multiplica- 
tions, as  in  arithmetic,  and  the  literal  factors  by  repeated 
additions  of  the  exponents. 

Thus,  the  3d  power  of  3  is  3x3x3=27,  but  the  3d  power  of 
a2  is  a2><ci2xa2=a-'+2+2^a2X3^af.. 

If  the  quantity  to  be  involved  is  positive,  any  power  of 
that  quantity  will  be  positive.  If  it  is  negative,  the  even 
powers  will  be  positive  and  the  odd  powers  negative. 

Thus,  — ax— «=+<^".  »nd  — aX— «X— «=— «^-  The  4th 
power  of  — a  is  +«'*;  the  5th  power  is  — a^;  and  so  on.  Hence, 
we  have  the  following 

Rule  for  Involving  a  Monomial. — 1.  Involve  the  coeffi- 
cient hy  the  rule  of  arithmetic. 

2.  Multiply  the  exponents  of  the  literal  factors  hy  the  ex- 
ponent of  the  required  power. 


130  RAY'S  ALGEBRA,  SECOND  BOOK. 

3.  If  the  quantity  he  negative^  make  the  even  powers  jpositive 
and  the  odd  powers  negative. 

1.  Find  the  square  of  bax^z^ Ans.  2ba^x*^. 

2.  The  square  of  —W'cd Ans.  9b*c'd\ 

3.  The  cube  of  2x'z Ans.  8xV. 

4.  The  cube  of  — 3aV Ans.  —2laW 

5.  The  fourth  power  of  — 2a:z'^.       .     .     .  Ans.  16ccV. 

6.  The  fifth  power  of  —Sd'b\   .     .    Ans.  —24Sa'%'\ 
Y.  The  seventh  power  of  — m'^n.     .     .      Ans.  — m^*«^ 

8.  The  square  of  a'^b^" Ans.  a^"'b*''. 

9.  The  nth  power  of  xy'^z^ Ans.  icy"^"^. 

10.  The  square  and  the  cube  of  |a'a;'"+y-^ 

(1)  Ans.  ^\a^x'"^+yp-\  (2)  -jf^aV'+y^-'. 

GX  (2  OC 

11.  The  square  of  .p-, Ans.  - — -. 

*  bz^  b'^z* 

12.  The  cube  of  g| ^''^-  2"^- 


Case  II.— To  square  a  Binomial  Quantity. 
The  rule  for  this  has  already  been  given,  Arts.  *78  and  79. 


1.  Find  the  square  a — x. 

2.  The  square  of  x-\-7/. 

3.  The  square  of  mx — nx^ 

4.  The  square  of  |a-f -^i. 

5.  The  square  of  ^~^  ^ 


.     .     .  Ans.  a^ — 2ax-\-x\ 

,     .     .  Ans.  x'^-\-2xi/-\-i/^. 

Ans.  m^ic^ — 2mnx^-\-7i^x*. 

.Ans.  „4.a2_|_|a6-^J62 

^^^  a?+104^H-25// 


m^ — n^  7n* — zmhi^-\-u* 

A  quantity,  consisting  of  three  or  four  terms,  may  be 
squared  on  the  same  principle,  by  reducing  it  to  the  form 
of  a  binomial,  squaring,  and  completing  the  operations  in- 
dicated. 

Thus,  a^b—C=a-\-(b—C).  Squaring,  we  have  a^-\-2a{b—c) 
_j_(6_c)2=a2_^2a6  -2ac+b^—2bc-\-c^. 

a-^b—c-\-d=z(a-^b)—(c—d).  Squaring,  (a-f6)2_2(a+6)(c—(i) 
.^(c— cf)2=a2-f2a6+62— 2ac— 26c+2arf+26d+c2— 2cd+d2. 


FORMATION  OF  POWERS.  131 

1  12 

6.  Find  the  square  of  cc —  —1.    A.  x'^—2x-{ — ^-\ 1. 


Case  III.— To  raise  a  Binomial  to  the  Third  Power. 

By  trial,  we  find  the  cube  of  a-\-b  to  be  a^-{-Sa^h-]-Sah^ 
-\-h^.     Hence, 

The  cube  of  a  binomial  is  equal  to  the  cube  of  the  frst 
term,  plus  three  times  the  square  of  the  first  info  the  second, 
plus  three  times  the  first  into  the  square  of  the  second,  plus 
the  cube  of  the  second. 

If  the  quantity  is  a  residual,  as  a — b,  the  result  will  be 
the  same,  except  that  the  signs  will  be  alternately  plus  and 
minus.  A  quantity  consisting  of  three  or  four  terms  may 
be  cubed  in  the  same  manner,  by  reducing  it  to  the  form 
of  a  binomial,  as  explained  above  in  Case  II. 

Thus,  (a-6-fc)3=[(a— 6)-f-c]3=(a— 6)3+3(a— 6)2c-f3(a— 6) 
C^-j-c^,  which  last  may  be  further  expanded. 

(a+6_c+d)3^[(a-{-6)— (c— d)]3=^(a+6)3— 3(a-f6)2(c— d)+, 
etc. 

1.  Find  the  cube  of  x-\-i/.  Ans,  x^-\-Sx'^i/-\-Sxi/'^-\-7^. 

2.  The  cube  of  2x—z.  Ans.  Sa^—12xh-^6xz'—^. 

3.  The  cube  of  3x-}-2i/.  Ans.  27ay'-{-^4x'i/-\-S6xf-\-Sf. 

/I         mi  1        /.    ^ — ^  *  ^^' — Sm^n-\-Smn^ — n^ 

4.  The  cube  of p^.       Ans. 


m — 2n  '  m^ — 6m'^n-\-12mn'^ — Sn^' 

5.  The  cube  of  la— lb.        Ans.  la^—l,a'b-^lab'—l^b\ 

6.  Involve  (cc Y.    A.  x^ — Sx-\ i=x^ 5— S(a: ). 

X  XT?  X?  X 

7.  Involve  (e^-f-e-^)». 

Ans.  e3^-f3e^-}-3e-'+e-''rrre3^-fe-3^-|-3(e^+e-'). 

8.  Involve  (x-\-y-\-zy. 

A.  aj'+3x2^+3x-^2+3a:/+6x^2+3a:22+/-f3/2-|-33^2^-f2*. 


132  R/VY'S  ALGEBRA,  SECOND  BOOK. 


Case  IV.— To  raise  a  Binomial  to  any  Power. 

Rules  for  raising  a  binomial,  or  residual  quantity,  to 
the  4th,  5th,  6th,  or  to  any  higher  power,  may  be  formed 
on  the  same  principle  as  those  given  under  Case  II  (See 
Theorems  I  and  II,  Art.  78)  and  Case  III.  An  easier 
method,  however,  was  discovered  by  Sir  Isaac  Newton, 
which  we  now  proceed  to  explain. 


NEWTON'S    THEOREM. 

Let  a-\-h  be  raised  to  the  sixth  power  by  actual  multi- 
plication. 

a  +  6 

a  4-  6 


a^-\-  a  b 

a^-\-2,a  6-[-     b- :  second  power  of  a+6,  or  (a+6)2. 

a  4-  b 


a3+2a26+    a  62 
+  a^b^  2a  62+     6^ 

a3-(-3a25_|_  3a  62-j-     63    .    .    .     third  power  of  a-j-6,  or  (a+6)3. 
a  +6 

a4+3a36+  3a262+    a  b^ 
+  a36+  3a262+  3a  63-f  6< 


a4^4a35_|_  6a262-|-  4a  63-f  6*    ....  fourth  power,  or  (a-f-6)<. 

a  -1-6 

a5-|-4a<6-|-  6a362-f  4a2634-  a  6* 
+  a46-^  4a362-^  6a263+  4a  64+65 

a5j-5a464-10a362+10a263-[-  5a  6^+6^ («+^)^- 

a  -f-6 


a6-f5a'^6+10a^624-10a363+  5a264-fa6^ 

+  a'^64-  5a4624-10a363-^10a26*+5a6->4-^'"' 
aH6a'^^+15a^6^+20a363+15a264+6a65+66    ....  (a+6)^ 


FORMATION  OF  POWERS.  133 

If  we  involve  a — 6,  the  result  will  be  the  same,  except 
that  the  signs  of  the  terms  will  he  alternately  plus  and  minus. 

The  above  results  exhibit  certain  uniform  laws  of  de- 
velopment, following  which  we  may  raise  a  binomial  to  any 
required  power  without  the  tedious  process  of  multiplica- 
tion.    These  laws  are  as  follows : 

1st.  Number  of  Terms. —  The  mimher  of  terms  in  any 
power  of  a  binomial  is  one  more  than  the  exponent  of  the 
power. 

Thus,  the  2d  power  has  3  terms,  the  3d  power  4  terms,  etc. 

2d.  Signs  of  Terms. — If  both  terms  of  the  binomial  are 
positive,  all  the  terms  will  be  positive. 

If  the  second  term  is  negative,  the  Ist,  3c?,  etc.,  or  the  ODD 
terms,  will  be  positive,  and  the  even  terms  negative. 

3d.  Exponents. —  TTie  exponent  of  the  leading  letter 
is  the  same,  in  the  first  term,  as  the  power  to  which  the  quan- 
tity is  to  be  raised,  and  diminishes  by  unify,  in  the  succeeding 
terms,  disappearing  in  the  last. 

The  FOLLOWING  letter  is  not  found  in  the  first  term,  but 
enters  the  second  with  an  exponent  of  one,  which  exponent 
increases,  by  unity,  in  the  succeeding  terms,  until  it  equals,  in 
the  last  term,  the  exponent  of  the  power. 

Thus,  {a^bf=a^-\-oJ'b^a^b--\-a''b^-\-a-b^-{-ab^-\'b^,  omitting 
coefficients. 

4th.  The  Coefficients. —  The  coefficient  of  the  first  and  last 
terms  is  always  unity ;  that  of  the  second  term  is  the  same  as 
the  exponent  of  the  LEADING  LETTER  in  the  first  term. 

The  coefficient  of  any  other  term  may  be  found  by  the 
following  rule  : 

Multiply  the  coefficient  of  any  term  by  the  exponent  of  its 
leading  letter,  and  divide  the  product  by  the  number,  express- 
ing the  place  of  that  term  in  the  series  for  the  coefficient  of 
the  succeeding  term. 


134  RAYS  ALGEBRA,  SECOND  BOOK. 

The  coefficients  of  all  terms  eqiialhj  distant  from  the  frst 
and  last  are  equal. 

In  the  application  of  this  theorem,  we  may  first  write  the 
literal  factors  alone,  and  afterward  supply  the  coefficients, 
according  to  the  rules  above  given,  or,  we  may  carry  for- 
ward both  operations  at  the  same  time.     Thus, 

Let  it  be  required  to  raise  x-\-y  to  the  Tth  power,  or  to 
expand  (x-\~yy. 

Literal  factors,  oc^ ,  x^y,  x^y-^  x^y^^  x^y^,  x'^y\  xif^  y'^. 
The  cotificient  of  the  1st  term  is  uuity;  .-.  1st  term  is  iC^. 
"  "  "     2d      "    "    7  "  2d      "     "  Ix^'y. 

u  u  u        3j         u      u     7X6       a  gj         u      u  21x^y\ 

u  a  .t        4th       «      «      21X-^    u   4th        a      u   35a:4^3 

Continuing  thus,  we  have  for  the  complete  expansion, 

rr7-f72;*^2/+21a;'''2/2+35a:V4-35xV-f- 213:2^/^-1- 7a:?/*' +2/^. 

As  a  second  example  by  the  other  method, 

Let  it  be  required  to  expand  (« — hy. 

The  first  term  will  be  a'"';  the  second,  6a^'b.  For  the  third,  multi- 
ply 6  by  5,  and  divide  the  product  by  2,  for  the  coefficient,  and 
annex  the  literal  factors.  This  gives  15a^b^.  Multiplying  15  by  4 
and  dividing  by  3,  we  have  for  the  next  term  20a^b^. 

Continuing  this  process,  we  find  the  next  term  to  be  15a^b^,  the 
next  6a6'*,  and  the  last  ^^     Giving  the  proper  signs,  we  have 
af>_6a'^6-[-15a^62_20a363+15a26»_6a6^-(-66. 

The  following  additional  facts  may  be  noted,  and  may 
serve  to  render  the  application  of  the  above  principles  stiH 
more  simple  : 

1st.  The  sum  of  the  exponents  in  every  term  is  the  same, 
and  is  always  equal  to  the  power  of  the  binomial. 

Thus,  in  the  first  of  the  above  examples,  the  sum  of  the  exponents 
in  every  term  is  7;  in  the  second  their  sum  is  6. 

2d.  If  the  power  of  the  binomial  be  even,  the  number 
of  terms  will  be  odd ;  but  if  the  power  be  odd,  the  number 


FORMATION  OF  TOWERS.  135 

of  terms  will  be  even.  In  the  former  case,  there  will  be 
one  middle  term,  and  in  the  latter  two,  to  the  left  and  right 
of  which  the  coefficients  are  the  same. 

Thus,  in  the  above  examples,  the  coefficients  are — 

For  the  6th  power,     1,     6,     15,     20,     15,     6,     1. 
For  the  7th  power,     1,     7,     21,     35,     35,     21,     7,     1. 

3d.  The  smn  of  the  coefficients,  in  every  case,  is  equal 
to  2  raised  to  the  required  power  of  the  binomial. 

Thus,  in  the  above  examples,  1-f  6+15H-20-}-15+6+l=64^2«, 

and  l4-7+21+35-f35+21+7+l:^128.=27. 

Expand  (a-{-hy.  .  .  .  Ans.  a'-\-4:a^h-{-Qa'h^^4ah^-^hK 
Expand  (x-\-i/y. 

Ans.  a:«+6a:^-fl5a:y-f20ccy-fl5a;y-|-6.T/-f/. 
Expand  («— a-)^  A.  a^— 5a*x-f  lOa^x'— 10aV+5aa;*— a:\ 
Expand  (a-f-cc)^. 

Ans.    a«4-8a^T-f28aV+56aV-f70aV+56aV+ 
28aV-f8ax^H-x8. 
Expand  (a—iy. 

Ans.    a^—9a%-\-B6a-'h'—S4a^h'-^126a'h'—12Qa'¥ 

If  one  or  both  of  the  terms  of  the  binomial  have  a  coeffi- 
cient or  exponent  greater  than  unity,  or  more  than  one 
literal  factor,  the  expansion  may  be  made  in  the  same  way, 
after  which  the  operations  indicated  must  be  completed. 

Thus,  {2x^-{-5a^y={2x^y-^A{2x^)^{5a^)-\-6{2x^y{5a^Y-\~4{2x^) 
(5a2)3+(5a2)43^16a:i2_f_i60a:^a2-f600a:''a*+1000a:3a6+625a«. 

Or,  put  m=2x^  and  n=5a^.   Then,  (2a;3-|-6a2)4=(m-f  ti)^.   Then, 
(w-f  n)'<=m'* +4m3n-(- 67n2n2-f  4mn3-f  n*. 

Returning  to  the  values  of  m  and  n,  we  have  m*=(2x^y=16x^\ 
4m^n  =4x(2rr3)3x  5a2    ^AxSx^X     ^^^=  160a;%2. 
6m2n2z=6X(2a:3)2x;5a2)2^6x4a;6x  25a4=  600x^aK 
4m  n^=4x  2x^   X^5a2)3^4y2a:3xl25a6z=1000a;3a6. 
n4=(5a2)4=625as. 

Hence,  {2x^-{ba^y^lQx^-^  lQ0x^a^-\-600x^a*-j-1000x^a^-]-625a». 


136  RAY  S  ALGEBRA,  SECOND  BOOK. 

In  a  similar  manner,  a  quantity  consisting  of  three  or 
four  terms  may  be  involved,  by  first  reducing  it  to  the 
form  of  a  binomial,  as  explained  in  Cases  II  and  III. 

1.  Raise  a;2_^3y  to  the  fifth  power,  or  expand  (ic^-j-Sy^. 
Ans.  a:'»-fl5a^y+90a;y-j-270a:y-f405xy-f243y''. 

2.  Expand  (2a''-^axy.       Ans.  Sa^-{-12a^x-}-6a*x^-\-a^a^, 
S.  Expand  (2a~\-Sxy. 

Ans.  16a*-\-96a'x-\-216aW-{-216ax'-\-Slx^. 
4:.  Expand  (^a—Sh)\ 

Ans.  Jga*— |«36-f2^7a'^62_54a63-f 81Z/*. 

5.  Cube  a+2Z>— c. 

Ans.  a^-\-6a'h—Sa'c-{- Sh'-\- 1 2a¥—l 2h'c—c'J^ Sac" 
-\-6hc'—12ahc. 

6.  Expand  (a-^-b-^c—dy. 

Ans.  a*-\-4a^h-\-  (ja'h^  +  4ah^  +  h*-}-  4a'c  +  12a'6c 
-\-12ah'c-{-4:b'c  —  4a^d—12a'hd—12ah'd—Wd 
-\-6a'c'-\-12abc'-]-6b'c'~12a'cd—24:abcd—12b'cd 
-^6a'd'-\-12abd'-{-6b-'d'-\-4ac^  —  12acM-\-12acd^ 
—4ad^-\-Uc^  —  12bc''d-\-12bcd'—Ud'-{-c*—4:c^d 

-j-echP—4:Cd^-^d\ 

In  many  cases,  as  in  some  of  the  examples  above  given,  it  will 
sometimes  be  found  most  convenient  to  involve,  by  repeated  multi- 
plicatioas,  under  the  general  rule. 

For  further  exercise,  take  the  following : 

1.  If  x-\-~=p,  show  that  x^-\ — ^=p^ — 3/?. 

2.  If  two  numbers  difi"er  by  unity,  prove  that  the  dif- 
ference of  their  squares  is  equal  to  the  sum  of  the  num- 
bers. 

3.  Show  that  the  sum  of  the  cubes  of  any  three  con- 
secutive integral  numbers  is  divisible  by  the  sum  of  those 
numbers. 

Note. — For  a  more  general  discussion  of  the  Binomial  Theorem, 
Bee  Art.  310. 


EXTRACTION  OF  THE  SQUARE  ROOT.  137 

II.    EXTRACTION    OF    THE    SQUARE   ROOT. 
EXTRACTION    OF    THE    SQUARE    ROOT    OF    NUMBERS, 

173.  The  Root  of  a  number  is  a  factor  which  multi- 
plied by  itself  a  certain  number  of  times  will  produce  the 
given  number. 

The  Second  or  Square  Root  of  a  number,  is  that  num- 
ber which  multiplied  by  itself;  that  is,  taken  twice  as  a 
factor,  will  produce  the  given  number. 

The  Extraction  of  the  Square  Root  is  the  process  of 
finding  the  second  root  of  a  given  number. 

174.  To  show  the  relation  that  exists  between  the  num- 
ber of  figures  in  any  given  number,  and  the  number  of 
figures  in  its  square  root,  take  the  first  ten  numbers  and 
their  squares : 

1,     2,     3,       4,       5,       6,       7,       8,       9,       10; 
1,    4,    9,    16,    25,     36,    49,    64,    81,    100. 

The  numbers  in  the  first  line  are  also  the  square  roots 
of  the  numbers  in  the  second. 

We  see  from  this,  that  the  square  root  of  1  is  1,  and 
the  square  root  of  any  number  less  than  100  is  either  one 
figure,  or  one  figure  and  a  fraction.     Hence, 

When  the  number  of  places  of  figures  in  a  number  is  not 
more  than  TWO,  the  number  of  places  of  figures  in  the  square 
root  will  be  ONE. 

The  square  root  of  100  is  10 ;  and  of  any  number 
greater  than  100  and  less  than  10000,  the  square  root 
will  be  less  than  100  ;  that  is,  it  will  consist  of  two  places 
of  figures.     Hence, 

When  the  number  of  places  of  figures  is  more  than  TWO, 
and  not  more  than  FOUR,  the  number  of  places  of  figures  in 
the  square  root  will  be  TWO. 
2d  Bk.  12 


138  RAY'S  ALGEBRA,  SECOND  BOOK. 

In  the  same  manner  it  may  be  shown,  that  when  tlie 
number  of  places  of  figures  is  more  than  /owr,  and  not 
more  than  six,  the  number  of  places  in  the  square  root 
will  be  three,  and  so  on. 

1T5.  Every  number  may  be  regarded  as  being  com- 
posed of  tens  and  units. 

Thus,  76  consists  of  7  tens  and  6  units;  and  576  consists  of  57  tens 
and  6  units.  Therefore,  if  we  represent  the  tens  by  t,  and  the  units 
by  u,  any  number  will  be  represented  by  t-{-u,  and  its  square  by 
the  square  of  t-\-u,  or  {t-\'U)^. 

{t-\-uf=t^-^2tu-\-u^=.t2-\-  {2t-]-u)u.     Hence, 

7%e  square  of  any  number  is  composed  of  two  quantities, 
one  of  which  is  the  square  of  the  tens,  and  the  other  twice  the 
tens  plus  the  units  multiplied  hy  the  units. 

Thus,  the  square  of  25,  which  is  equal  to  2  tens  and  5  units,  is 

2  tens  squared  ^(20)2  =  400 
(4  tens  +  5  units)  multiplied  by  5^(40+5)5=225 

~625 
1.  Required  to  extract  the  square  root  of  625. 

Since  the  number  consists  of  three  places  625|25 

of  figures,  its  root  will  consist  of  two  places,  400| 

according    to   the   principle   established    in  20x2=40  225 

Art.  174,  we,  therefore,  separate  it  into  two  5  225 

periods,  as  in  the  margin.  45 

Since  the  square  of  2  tens  is  400,  and  of  3  tens  is  900,  it  is 
evident  that  the  greatest  square  contained  in  600  is  the  square 
of  2  tens  (20) ;  the  square  of  2  tens  (20)  is  400.  Subtracting  this 
from  625,  the  remainder  is  225. 

The  remainder,  225,  consists  of  twice  the  tens  plus  the  units, 
multiplied  by  the  units;  that  is,  by  the  formula,  it  is  {2t-\-u)u^  of 
which  t  is  already  found  to  be  2,  and  it  remains  to  find  u. 

Now,  the  product  of  the  tens  by  the  units  can  not  give  a  product 
less  than  tens;  therefore,  the  unit's  figure  (5)  forms  no  part  of  the 
double  product  of  the  tens  by  the  units.  Hence,  if  we  divide  the 
remaining  figures  (22)  by  the  double  of  the  tens  (4),  the  quotient 
will  be  the  unit's  figure,  or  a  figure  greater  than  it. 


EXTK ACTION  OF  THE  SQUARE  ROOT.      139 

Dividing  22  by  4  (2^)  gives  5  (u)  for  a  quotient.  This  unit's 
figure  (5)  is  to  be  added  to  the  double  of  the  tens  (40),  and  the  sum 
multiplied  by  the  unit's  figure. 

Multiplying  40+5==45(2^-f  w),  by  5  (u),  the  product  is  225,  which 
is  double  the  tens  plus  the  units,  multiplied  by  the  units.  As  there 
is  no  remainder,  we  conclude  that  25  is  the  exact  square  root 
of  625. 

In  squaring  and  doubling  the  tens,  it  is  customary  625|25 

to  omit  the  ciphers,  and  to  add  the  unit's  figure  to  400 

the  double  of  the  tens,  by  merely  writing  it  in  the  451225 

unit's  place.      The  actual  operation   is   usually  per-  |225 

formed  as  in  the  margin. 

2.  Required  to  extract  the  square  root  of  59049. 

Since  this  number  consists  of  five  places  of  59049|243 

figures,  its    square   root   will   consist  of  three  4 

places.     (Art.  174.)     We,  therefore,  separate  it  441190 
into  three  periods.  |l76 

In    performing   this    operation,  we   find  the  488 1 1449 
square   root    of  the  number  590,  on  the  same  11449 

principle  as  in  the  preceding  example. 

We  next  consider  24  as  so  many  tens,  and  proceed  to  find  the 
unit's  figure  (3)  as  in  the  preceding  example. 

From  these  illustrations,  we  derive  the  following 

Rule  for  the  Extraction  of  the  Square  Root  of  Num- 
bers.— 1st.  Separate  the  given  number  into  periods  of  two 
places  each,  beginning  at  the  unit's  place.  (The  left  period 
will  often  contain  but  one  figure.) 

2d.  Fi7id  the  greatest  square  in  the  left  period,  and  place 
its  root  on  the  right,  after  the  manner  of  a  quotient  in  divi- 
sion. Subtract  the  square  of  the  root  from  the  left  period, 
and  to  the  remainder  bring  down  the  next  period  for  a  divi- 
dend. 

3d.  Double  the  root  already  found,  and  place  it  on  the 
left  for  a  divisor.  Find  how  many  times  the  divisor  is  con- 
tained in  the  dividend,  exclusive  of  the  right  hand  figure,  and 
place  the  figure  in  the  root  and  also  on  the  right  of  the 
divisor. 


140  RAYS  ALGEBRA,  SECOND  BOOK. 

4th.  Multiply  the  divisor  thus  increased  hy  the  last  figure 
of  the  root;  subtract  the  product  from  the  dividend^  and  to 
the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend. 

5th.  Double  the  whole  root  already  found  for  a  new  divi- 
sor^ and  continue  the  operation  as  before,  until  all  the  periods 
are  brought  down. 

Note  — If,  in  any  case,  the  division  can  not  be  eifected,  place  a 
cipher  in  the  root  and  divisor,  and  bring  down  the  next  period. 

ITG.  In  extracting  the  square  root  of  numbers,  the  re- 
mainder may  sometimes  be  greater  than  the  divisor,  while 
the  last  figure  of  the  root  can  not  be  increased.  To  ex- 
plain this, 

Let  a  and  a-|-l,  be  two  consecutive  numbers. 

Then,  {a-f^lY^=a'^-\-2aA~l,    the  square  of  the  greater. 
And         (a)2=a2^  "        "  "      less. 

Their  difference  is    2a-f  1.     Hence, 

TJie  difference  ef  the  squares  of  two  consecutive  numbers  is 
equal  to  twice  the  less  number,  increased  by  unify. 

Therefore,  when  atiy  remainder  is  less  than  twice  the 
root  already  found,  plus  one,  the  last  figure  can  not  be  in- 
creased. 

Required  the  square  root  of 

1.  2601.  .  .  .  Ans.  51.  15.  43046Y21.   Ans.  6561. 
2.7225.  .  .  .  Ans.  85.  16.  49042009.   Ans.  7008. 

1061326084.  A.  32578. 


3.  47089.  .  .   Ans.  217. 

4.  390625.  .  Ans.  625. 


8.  948042681.  Ans.  30709. 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS. 

177.  Since  |X|=ifV>  t^e  square  root  of  rj%  is  |;  that 

i/4 
is,  1/25=  Toe "-=|-     Hence,  we  have  the  following 


EXTRACTION  OF  THE  SQUARE  ROOT. 


141 


Rule  for  Extracting  the  Square  Root  of  a  Fraction. — 

SJxtract  the  square  root  of  both  terms. 

When  the  terms  of  a  fraction  are  not  perfect  squares,  they  may 
sometimes  be  made  so  by  reducing.     Thus, 

Find  the  square  root  of  ff. 


Here,  J-^ 


4X5 


9X.^ 


By  canceling  the  common  factor  5,  the  fraction 


becomes  |,  the  square  root  of  which  is  |. 

When  both  turms  are  perfect  squares,  and  contain  a  common  fac- 
tor, the  reduction  may  be  made  either  before  or  after  the  square 
root  is  extracted. 

Thus,  v/if-|=i;  or,  M=|,  and  i/4_f. 


Required  the  square  root  of 


fi4 

T2T- 
•2  2  5 
400- 


Ans.  |. 


Q       9  74  7  Ans     '^^ 

^-  T0092-        •  •        '^°^-    oS- 

A         5  6169  Ans      -37 

^'  TOOOOOO-  ^^^'  1000' 


ITS.  A  Perfect  Square  is  a  number  whose  square  root 
can  be  exactly  ascertained  ;  as,  4,   9,  16,  etc. 

An  Imperfect  Square  is  a  number  whose  square  root 
can  not  be  exactly  ascertained ;  as,  2,  3,  5,  6,  etc. 

Since  the  difference  of  two  consecutive  square  num- 
bers, a^  and  a^-|-2a-f  1,  is  2a-|-l  ;  therefore,  there  are 
always  2a  imperfect  squares  between  them. 

Thus,  between  the  square  of  5  (25)  and  the  square  of  6  (36),  there 
are  10  {2a=i2y^5)  imperfect  squares. 

A  quantity,  affected  by  a  radical  sign,  whose  root  can 
not  be  exactly  found,  is  called  a  radical.,  or  surd,  or  irra- 
tional root;   as,  \/2,  i^5,  etc. 

The  root  of  such  a  quantity,  expressed  with  more  or 
less  accuracy  in  decimals,  is  called  the  approximate  value, 
or  approximate  root.  Thus,  1.414-|-  is  the  approximate 
value  of  |/2. 

lTO«  It  might  be  supposed,  that  when  the  square  root  of  a 
whole  number  can  not  be  expressed  by  a  whole  number,  it  might  be 
exactly  equal  to  some  fraction.    That  it  can  not,  will  now  be  shown. 


142  RAY'S  ALGEBRA,  SECOND  BOOK. 

Let  c  be  an  imperfect  square,  as  2,  and,  if  possible,  let  its  square 
root  be  a  fraction,  -j,  in  its  lowest  terms. 

Then,   |/c=-t;  and  C=^,  by  squaring  both  sides  (Art.  148). 

Now,  by  supposition,  a  and  b  have  no  common  factor;  therefore, 
their  squares,  a^  and  6^,  can  have  no  common  factor,  since  to 
square  a  number,  we  merely  repeat  its  factors.  Consequently, 
a2 

^„  must  be  in  its  lowest  terms,  and  can  not  be  equal  to  a  whole 
^-  a^ 

number.     Hence,  the  equation  C=^,  is  not  true,  and  the  supposi- 

(X 
Hon  on  which  it  is  founded,  that  is,  that  -^0^=^-,  is  false;    there- 
fore, the  square  root  of  an  imperfect  square  can  not  be  a  fraction. 


APPROXIMATE    SQUARE    ROOTS. 

ISO.  To  explain  the  method  of  finding  the  approxi- 
mate square  root  of  an  imperfect  square,  let  it  be  required 
to  find  the  square  root  of  5  to  within  J. 

If  we  reduce  5  to  a  fraction  whose  denominator  is  9  (the  square 
of  3,  the  denominator  of  the  fraction  1),  we  have  5='*^^. 

Now,  the  square  root  of  ^^  is  greater  than  |,  and  less  than  |; 
hence,  |,  or  2,  is  the  square  root  of  5  to  within  1. 

To  generalize  this  explanation,  let  it  be  required  to  ex- 
tract the  square  root  of  a  to  within  a  fraction  -. 

Write  a  (Art.  127)  under  the  form  — ^,  and  denote  the  entire 
part  of  the  square  root  of  an^  by  r.    Then,  an^  will  be  comprised 

between  r^  and  (r-|-l)2,  and  the  square  root  of  — y  ^^^i  he  comprised 

r       ,  r+l  ^ 

between  —  and   . 

n  n 

V  7*4-1  1  T 

But  the  difference  between-  and  — i—  is  — :    therefore,  —  rep- 
n  n         n^  ^  n      ^ 

resents  the  square  root  of  a  to  within  -.     Hence, 


EXTRACTION  OF  THE  SQUARE  ROOT.       143 

Rule  for  Extracting  the  Square  Root  of  a  Whole  Num- 
ber to  within  a  Given  Fraction. — 1.  Multiply  the  given 
number  hy  the  square  of  the  denominator  of  the  fraction, 
which  determines  the  degree  of  approximation. 

2.  Extract  the  square  root  of  this  product  to  the  nearest 
unit  J  and  divide  the  residt  hy  the  denominator  of  the  fraction. 

1.  Find  the  square  root  of  3  to  within  J.     .     Ans.  1|. 

2.  Of  10  to  within  -J Ans.  3. 

3.  Of  19  to  within  J Ans.  4J. 

4.  Of  30  to  within  j\ Ans.  5.4. 

6.  Of  75  to  within  -^Jg Ans.  8.66. 


Since  the  square  of  10  is  100,  the  square  of  100, 10000, 
and  so  on,  the  number  of  ciphers  in  the  denominator  of  a 
decimal  fraction  is  doubled  by  squaring  it.     Therefore, 

WJien  the  fraction  which  determines  the  degree  of  approxi- 
mation is  a  decimal,  add  two  ciphers  for  each  decimal  place 
required;  and,  after  extracting  the  square  root,  point  off  from 
the  right  one  place  of  decimals  for  each  two  ciphers  added. 

6.  Find  the  square  root  of  3  to  five  places  of  decimals. 

Ans.  1.73205. 

7.  Find  the  square  root  of  7  to  five  places  of  decimals. 

Ans.  2.64575. 

8.  Find  the  square  root  of  500.    Ans.  22.360679+. 

181.  To  find  the  approximate  square  root  of  a  fraction. 


1.  Required  to  find  the  square  root  of  ^  to  within 


4  — 4\/I— 2f 


The  square  root  3|  is  greater  than  ^  and  less  than  ^;  therefore, 
^  is  the  square  root  of  4  to  within  less  than  ^.  Hence,  to  find  the 
square  root  of  a  fraction  to  within  one  of  its  equal  parts, 

Rule. — Multiply  the  numerator  by  its  denominator,  extract 
the  square  root  of  the  product  to  the  nearest  unit,  and  divide 
the  result  by  the  denominator. 


144  RAY'S  ALGEBRA,  SECOND  BOOK. 


2.  Find  the  square  root  of  {'^  to  within  j\.         Ans. 


3.  Find  the  square  root  of  A  J  ^o  within  -^Jq.         Ans.  -f^. 

It  is  obvious  that  any  decimal,  or  whole  number  and  decimal, 
may  be  written  in  the  form  of  a  common  fraction,  and  having  its 
denominator  a  perfect  square,  by  adding  ciphers  to  both  terms. 
Thus,  .S=j%^j%%;  .156=Vo'o%;  1.2=.lgg,  and  so  on. 

Therefore,  to  extract  the  square  root,  as  in  the  method  for  the 
approximate  square  root  of  a  whole  number  (Art.  180), 

Rule. — 1.  Annex  ciphers  to  the  decimal,  until  the  number 
of  decimal  places  shall  be  equal  to  double  the  number  required 
in  the  root. 

2.  After  extracting  the  root,  point  off  from  the  right  the 
required  number  of  decimal  places. 

4.  Find  the  square  root  of  .4  to  six  places. 

Ans.  .632455+. 

5.  Find  the  square  root  of  7.532  to  five  places. 

Ans.  2.74444+. 

When  the  denominator  of  a  fraction  is  a  perfect  square,  extract 
the  square  root  of  the  numerator  to  as  many  places  of  decimals  as 
are  required,  and  divide  the  result  by  the  square  root  of  the 
denominator. 

Or,  reduce  the  fraction  to  a  decimal,  and  then  extract  its  square 
root.  When  the  denominator  of  the  fraction  is  not  a  perfect  square, 
the  latter  method  should  be  used. 

6.  Find  the  square  root  of  ^^g  to  five  places. 

V^5=:2.23606+,   i/16=4,    i/fg^^^'  ~  ^ ^  ^^-^=.55901+ 
Or,  -5g=.3125,  and  ;/ .3 125 =.55901+. 

7.  Find  the  square  root  of  |.       .  .     Ans.  .774596+. 

8.  Find  the  square  root  of  If    .  .     Ans.  1.11803+. 

9.  Find  the  square  root  of  3^.    .  .  Ans.  1.903943+. 
10.  Find  the  sqtiare  root  of  ll'^.  .  .  Ans.  3.349958+. 


EXTRACTION  OF  THE  SQUARE  ROOT.  145 


EXTRACTION    OF    THE    SQUARE    ROOT     OF     ALGE- 
BRAIC    QUANTITIES. 

EXTRACTION    OF    THE    SQUARE    ROOT    OF    MONOMIALS. 

18!3.  To  square  a  monomial,  (xlrt.  172),  we  square  its 
coefficient,  and  multiply  each  exponent  by  2. 

Thus,  (3mn2)2^9m2n4. 

Therefore,  ^9m^n^=Smn^.     Hence,  we  have  the  following 

Rule  for  Extracting  the  Square  Root  of  a  Monomial. — 

Extract  the  square  root  of  the  coefficient  as  a  niwiber,  and 
divide  the  exponent  of  each  letter  hy  '2i. 

Since  +aX-f  «=-|-«^,    — aX— «=+«^; 

Therefore,   ■^a^=-\-a,  or  —a.     Hence, 

The  square  root  of  any  positive  quantity  is  either  plus 
or  minus.  This  is  expressed  by  writing  the  double  sign 
before  the  root.  Thus,  y'4a^=r=h2a;  read,  plus  or 
minus  2a. 

If  a  monomial  is  negative,  the  extraction  of  the  square 
root  is  impossible,  since  the  square  of  any  quantity,  either 
positive  or  negative,  is  necessarily  positive.  Thus,  i/ — 4, 
-|/ — b,  are  algebraic  symbols,  which  indicate  impossible 
operations. 

Such  expressions  are  termed  imaginary  quantities.  In 
an  equation  of  the  second  degree,  they  often  indicate  some 
absurdity,  or  impossibility  in  the  equation  or  problem  from' 
which  it  was  derived. 

1.  IQxhj^.  Ans.  d=4.T?/^  I  3.  m^x^y^z^.       Ans.  -±zmxhf7}. 

2.  25m2n2.  Ans.  drSmn.  |  4.   1024a26V«.Ans.=b32aZ/V. 

/  a  \2     a     a    a^  la^    ^a^        a 

Since  I   -X  |=tXt-=7-t,;  therefore,  -x-j-^z^-^-^-z^ziz^.     Hence, 
\  b!      ly^b    b-'  '  V6-'    |/62         ^  ' 

To  find  the  square  root  of  a  monomial  fraction^  extract  the 
square  I'oot  of  both  terms. 
2d  Bk.  13* 


146  RAY'S  ALGEBRA,  SECOND  BOOK. 

5.  Find  the  square  root  of  — -.     .     .     .       Ans.  ± — ■. 

6.  Find  the  square  root  of  T7>~Tr.-      •     •     Ans.  rir=— ^. 


EXTRACTION  OF  THE  SQUARE  ROOT  OF  POLYNOMIALS. 

183.  In  order  to  deduce  a  rule  for .  extracting  the 
square  root  of  polynomials,  let  us  first  find  the  relation 
that  exists  between  the  several  terms  of  any  quantity  and 
its  square. 

(a-(-6+c)2=  a2-j-2a6-|-62-|-2ac+26c-f  e2  =ra2^(2a-(-6)6+(2a 
-f26+c)c. 

{a+6+c'+d)2  =:  rt2-f  2a6-f  62_^2ac+26c-f  c2-f-2a(^+26cZ+2crf 
-j-d2^«2^(2a+6)6+(2a+26+c)c+(2a+26+2c+d)o?. 

Or,  by  calling  the  successive  terms  of  a  polynomial,  r,  r^,  r^\  r'^\ 
and  so  on,  we  shall  have  (r+r^-|-r^''-[-r^^^)2=r2-j-(2r-]-7^)7^-f  (2r 
-f  2r^-f  r^^)r'^+(2r-f  2r^  +  2r'^+r''07^^^,  where  the  law  is  mani- 
fest. 

In  this  formula,  r,  7^,  r^'^  r'^^^  may  represent  any  algebraic  quan- 
tities whatever,  either  integral  or  fractional,  positive  or  negative. 

This  formula  gives  the  following  law .-, 

The  square  of  any  polynomial  is  equal  to  the  square  of  the 
first  term  —  plus  twice  the  first  term^  plus  the  second,  mul- 
tiplied hy  the  second  —  plus  twice  the  first  and  second  terms, 
plus  the  third,  multiplied  by  the  third  —  plus  twice  the  first, 
second,  and  third  terms,  plus  tjie  fourth,  multiplied  by  the 
fburth,  and  so  on. 

Hence,  by  reversing  tlie  operation,  we  have  the  following 

Rule  for  Extracting  the  Square  Root  of  a  Polyno- 
mial.— 1st.  Arrange  the  polynomial  with  reference  to  a  cer- 
tain letter. 

2d.  Extract  the  square  root  of  the  first  term,  place  the 
result  on  the  right,  and  subtract  its  square  from  the  given 
quantity. 


EXTRACTION  OF  THE  SQUARE  ROOT.      I47 

Sd.  Divide  the  first  term  of  the  remainder  hy  double  the 
part  of  the  root  already  found ^  and  annex  the  result  to  both 
the  root  and  the  divisor.  Multiply  the  divisor  thus  increased 
by  the  second  term  of  the  root,  and  subtract  the  product  from 
the  remainder. 

4th.  Double  the  terms  of  the  root  already  found  for  a 
partial  divisor,  divide  the  first  term  of  the  remainder  by  the 
first  term  of  the  divisor,  and  proceed  in  a  similar  manrier  to 
find  the  other  terms. 

1.  Find    the  square  root   of  4:x'^y'^-\-12x^y-\-9x'^ — SOxy^ 
Arranging  the  polynomials  with  reference  to  y,  we  have 

ROOT, 

2by^~20xy^-{-'^x^?/^—S0xy^-\-V2x^y-\-9x^\5y^^2xy-3x 
2V 

10?/2-  2xy\-20xy^-^Ax^y^ 
I  -  2()a:?/H4a;2y2 


10y^—ixy-3xl  -S0xy^^l2x^y-^9x^ 
\-S0xy2-]-12x^y^9x2 

If  the  preceding  example  be  arranged  according  to  the  powers 
of  X,  the  root  found  will  be  'Sx-{-2xy—5y^.  This  is  correct  also,  as 
may  be  shown  generally,  thus : 

^ {a^-\-2ax-^x^)^.db{a-\-x)=a-\-x,  or  —a — x. 

2.  x:'-^6ax-\-9a^ Ads.  cc-f  3a. 

3.  16x'^— 40a-y-f  25/ Ans.  4x—by. 

4.  4x''z'—12xyz-i^9y' Ans.  2xz—Sy. 

5.  49a*'"-«— 42a«'''-2-|-9a«'"+2.   .     .  Ans.  Va'^'"-'— 3a*'"+^ 

6.  l_|.2.T-[-7aj'-f  6a;^+9.x* Ans.  l-\-x-{-Sx\ 

7.  9a*—12a'b-\-S4:d'b'—20al/-{-2^b'. 

Ans.  Sa'—2ab-i-bb\ 

8.  x^-\-4:x'-\~10x*-Jr20x'-^2bx'-^24x-^16. 

Ans.  a;3+2x2+3a;+4. 

9.  9x'  —  6xy-{-S0xz-\-6xt-^y'  —  10yz  —  2yt-j-2bz'-^10zt 
-\-f.  Ans.  3x—y-{-bz-\-t. 


148  RAYS  ALGEBRA,  SECOND  BOOK. 


10.  x'—2x'^i^-^~'-.-{-J^ Ans.  :i;2— a;+i 


-^      2^a'b^      bahc^     c*  .         hah      & 

11.  -^ ^  +  ^ ^^«-2--8- 

12.  -25 y--^-+94-49a:^     Airs.  7a;^-^H-a 

Id.    77—2+- Ans. . 

b^  a?  b      a 

14.  Reduce  the  following  expression  to  its  simplest 
form,  and  extract  the  square  root : 

Qi—iy—2{a^^b''){a~by-\-2{a'^b').    .      .  Ans.  d'-\-b\ 

15.  Find  the  square  root  of  1 — x^  to  five  terms. 

.         ^      x^      x^       x^       hx^ 
Ans.  1- 2- g-jg-j28-'  ''"■ 

16.  Find  the  first  five  terms  of  the  square  root  oix^-^-a^. 

.  ,   a^         a*    ,      a^  5a^     , 

184.  The  following  remarks  will  be  found  useful : 
1st.  Ab  binomial  can  be  a  perfect  square;  for  the  square 
of  a  monomial  is  a  monomial,  and  the  square  of  a  bino- 
mial is  a  trinomial. 

Thus,  a2_j_^2  jg  jjQt  a  perfect  square,  but  if  we  add  to  it,  or  sub- 
tract from  it,  2a6,  it  becomes  the  square  of  a-j-6  or  of  a—b. 

2d.  In  order  that  a  trinomial  may  be  a  perfect  square, 
the  two  extreme  terms  must  be  perfect  squares,  and  the 
middle  term  double  the  product  of  the  square  roots  of  the 
extreme  terms. 

Hence,  to  find  the  square  root  of  a  trinomial  when  it  is 
a  perfect  square, 

Extract  the  square  roots  of  the  extreme  terms,  and  unite 
them  by  the  sign  of  the  second  term. 

Thus,  a^-\-4iax-\-Ax^  is  a  perfect  square,  and  its  square  root  is, 
a-\-2ac;  4x--{^8xy  ]-9y^  is  not  a  perfect  square.  For  other  illustra- 
tions, see  Exs.  2,  3,  4,  11,  and  13,  Art.  183. 


EXTRACTION  OF  THE  CUBE  ROOT.  149 

III.    EXTRACTION    OF    THE    CUBE    ROOT. 
EXTRACTION     OF     THE     CUBE     ROOT     OF     NUMBERS. 

185.  The  Cube,  or  third  power  of  a  number,  is  the 
product  arising  from  taking  it  three  times  as  a  factor. 
(Art.  172.) 

The  Cube  Hoot,  or  third  root  of  a  number,  is  one  of  three 
equal  factors  into  which  it  may  be  resolved. 

To  extract  the  cube  root  of  a  number,  is  to  find  a  num- 
ber which,  taken  three  times  as  a  factor,  will  produce  the 
given  number. 

186.  To  show  the  relation  that  exists  between  the 
number  of  figures  in  any  given  number,  and  the  number 
of  figures  in  its  cube  root. 

The  first  ten  numbers  and  their  cubes  are. 

Roots,     1,     2,       3,       4,       5,       6,       7,       8,       9,       10; 
Cubes,     1,     8,     27,     G4,  125,  216,  343,  512,  729,  1000. 

We  see  from  this  that  the  cube  of  a  number  consisting  of  one 
place  of  figures,  does  not  exceed  three  places. 

Again,  comparing  the  numbers  10  and  100,  we  have, 

Numbers,     ....     10,  100; 

Cubes, iOOO,  1*006006. 

Since  the  cube  of  10  is  1000,  and  the  cube  of  99,  which  is  less 
than  100,  is  less  than  1000000;  therefore,  the  cube  of  a  number 
consisting  of  two  places  of  figures,  has  more  than  three  places,  and 
not  more  than  six  places  of  figures. 

Again,  since  the  cube  of  100  is  IOO6OO6,  and  the  cube  of  1000  is 
1006000006;  the  cube  of  a  number  consisting  of  three  places  of 
figures  has  more  than  six  places,  and  not  more  than  nine  places  of 
figures. 

If,  therefore,  we  begin  at  the  unit's  place  of  a  number,  and  sepa- 
rate it  into  periods  of  three  places  each,  the  number  of  periods  will 
show  the  number  of  places  of  figures  in  the  root.  The  left  period 
will  often  contain  only  one  or  two  figures. 


150  RAY'S  ALGEBRA,  SECOND  BOOK. 

X8T-  To  investigate  a  rule  for  the  extraction  of  the 
cube  root. 

The  first  step  is  to  show  the  relation  that  exists  between 
any  number  composed  of  units  and  tens,  and  its  cube. 

Let     .     .     ^=  the  tens,  and  w=  the  units  of  a  given  number. 

Then,     t-\-U=  the  number. 

And  [t-\-u)^=  the  cube  of  the  number. 

But  (^+w)3.^^3^3^2^^3^i^2^^^3^j;3^^3^2^3^^^_|_^2)^,     HencB, 

77ie  cube  of  any  number  consisting  of  tens  and  units,  is 
equal  to  the  cube  of  the  tens,  —  plus  three  times  the  square 
of  the  tens,  plus  three  times  the  product  of  the  tens  and  mi  its, 
jilus  the  square  of  the  units,  all  three  multiplied  by  the 
units. 

1.  Required  to  extract  the  cube  root  of  13824. 

Separating  the  number  into  periods 
by  points,  we  find  there  will  be  two 
figures  in  the  root.  The  greatest  cube 
in  13  (thousand)  is  8  (thousand);  the 
cube  root  of  which  is  2  [t) ,  and  its 
cube,  8  (thousand),  corresponds  to  t^  in 
the  formula. 

We  then  subtract  this  from  the  given 
number,  and  find  a  remainder  5824,  which  corresponds  to  [St^-^Stu 
-^u'^)u  in  the  formula.  The  first  term,  St^,  of  this  formula,  is 
sometimes  termed  the  trial  divisor,  as  it  is  used  to  find  the  units 
figure  u. 

If  the  remaining  terms  were  only  St^u,  we  could  readily  find  u  by 
dividing  by  St^;  but  if  we  divide  by  3^2^  we  may  obtain  a  figure 
too  large,  on  account  of  omitting  the  terms  S(u-{^u'^,  of  which  u  is 
as  yet  unknown.  But  if  we  first  obtain  a  figure  too  large,  at  a  sec- 
ond trial  we  must  take  one  that  is  less. 

Since  the  square  of  tens  is  hundreds,  in  using  three  times  the 
square  of  the  ten's  figure  as  a  trial  divisor,  we  omit  the  figures  (24) 
in  the  unit's  and  ten's  places  of  the  dividend. 

In  this  case,  12  is  contained  in  58  four  times.  This  gives  4  (u)  for 
the  required  unit's  figure,  and  we  now  find  the  complete  divisor, 
3<2^3<w-|-w2^1200+240+16:z^l45e. 


tu 

13824|24 

8 

3^2  ^1200 

5824 

3tu^-  240 

u^=     16 

1456 

5824 

EXTRACTION  OF  THE  CUBE  ROOT. 


151 


Multiplying  this  by  4  (w),  the  product  is  5824,  which,  subtracted 
from  ilie  first  remainder,  leaves  zero  (0),  and  shows  that  24  is  the 
exact  cube  root  required. 

In  cubing  the  tens,  it  is  customary  to  omit  the  ciphers;  but  in 
taking  three  times  the  square  of  the  tens,  also  in  taking  three  times 
the  product  of  the  tens  by  the  units,  it  is  best  to  write  ciphers  in 
the  vacant  orders. 


2.  Required  to  find  the  cube  root  of  44361864. 

44361864i354 


After  separating  the  number 
into  periods,  we  find  the  cube 
root  (35)  of  44361  on  the  same 
principles  as  in  the  preceding 
example.  Then,  considering  35 
(lO/t-f-^)  as  so  many  tens,  we 
find  the  unit's  figure  (4),  as  in 
the  preceding  example. 

In  dividing  by  the  trial  divi- 
sor 27,  to  find  the  second  fig- 
ure (5),  we  first  obtain  6,  but 
this  is  found  by  trial  to  be  too 
large. 


27 

3/i2=2700  17361 
37i^=  450 
1'^=    25 

3175  15875 

3(7^+^)2=3675001486864 
3(/i+<)M=    4200 

W2=r  16 


3717161486864 


From  the  preceding,  we  derive  the  following 


Rule  for  the  Extraction  of  the  Cube  Root  of  Num- 
bers.— 1st.  Separate  the  given  number  into  j)eriods  of  three 
jylaces  each,  beginning  at  the  unit's  2}lace.  (The  left  period 
will  often  contain  but  one  or  two  figures.) 

2d.  Find  the  greatest  cube  in  the  left  period,  and  place  its 
root  on  the  right,  as  in  division.  Subtract  the  cube  of  the 
root  from  the  left  period,  and  to  the  remainder  bring  down 
the  7icxt  period  for  a  dividend. 

3d.  Square  the  root  already  found,  and  midtiply  it  by  3  for 
a  trial  divisor.  Find  how  many  times  this  divisor  is  con- 
tained in  the  dividend,  omitting  the  unit's  and  tens  figures, 
and  write  the  result  in  the  root.  Add  together,  the  trial  divi- 
sor with  two  ciphers  annexed ;  three  times  the  product  of  the 
last  figure  of  the  root  by  the  rest,  with  one  cipher  annexed; 


1.52  HAYS  ALGEBRA,  SECOND  BOOK. 

and  the  square  of  the  last  figure;  the  sum  will  he  the  complete 
divisor. 

4th.  Multiply  the  complete  divisor  hy  the  last  figure  of  the 
root,  and  subtract  the  product  from  the  dividend j  and  to  the 
remainder  bring  down  the  next  period  for  a  new  dividend, 
and  so  proceed  until  all  the  periods  are  brought  down. 

Extract  the  cube  root  of  the  followino:  numbers  : 


3.  12167.  .  .  Ans.  23. 

4.  39304.  .  .  Ans.  34. 

5.  493039..  .  Ans.  79. 

6.  2097152.  Ans.  128. 


7.  127263527.  Ans.  503. 

8.  403583419.  Ans.  739. 

9.  158252632929. 

Ans.  5409. 


By  a  process  of  reasoning  similar  to  that  given  in  Art.  177,  we 
deduce  the  following 

Rule  for  Extracting  the  Cube  Root  of  a  Fraction. — 

Reduce  the  fraction^   if  necessary,   to   its   lowest   terms,  and 
extract  the  cube  root  of  both  terms. 

10.  Find  the  cube  root  of  -^^^ Ans.  i. 

11.  Find  the  cube  root  of  ^f^'/^ Ans.  f. 

188.  A  Perfect  Cube  is  a  number  whose  cube  root  can 
be  exactly  ascertained  ;  as,  8,  27,  64,  etc. 

An  Imperfect  Cube  is  a  number  whose  cube  root  can 
not  be  exactly  ascertained  ;  as,  2,  3,  4,  etc. 

It  may  be  shown,  by  a  course  of  reasoning  precisely 
similar  to  that  employed  in  Art.  179,  that  the  cube  root 
of  an  imperfect  cube  can  not  be  a  fraction. 

APPROXIMATE    CUBE    ROOTS. 

189.  To  illustrate  the  method  of  finding  the  approxi- 
mate cube  root  of  an  imperfect  cube,  let  it  be  required 
to  find  the  cube  root  of  6  to  within  J.     6=-3gY-. 

Now,  the  cube  root  of  384  is  greater  than  7  and  less  than  8;  there- 
fore, the  cube  root  of  364  is  greater  than  |  and  less  than  |;  hence. 


is  the  cube  root  of  6  to  within  less  than  1 


EXTRACTION  OF  THE  CUBE  ROOT.        153 


root   of  a    number   a.   to   within   a  fraction    — , 

n 


To  generalize  this  method,  let  it  be  required  to  extract  the  cube 
within   a  fra 
_aX^"^ ^^^ 

Let  r  be  the  root  of  the  greatest  cube  contained  in  an^;  then, 
— 5-  IS  comprised  between  — ;r  and    — ;, —  :  hence,  its  cube  root  is 

r  7*4 1 

comprised  between  -   and  ;   and  since  the  difference  of  these 

n  n 

1  r  1 

fractions  is  — ;  therefore,  -  is  the  cube  root  of  a  to  within  — .     Hence, 


Rule  for  Extracting  the  Cube  Root  of  a  Whole  Num- 
ber to  within  a  Given  Fraction. — Multiply  the  given  num- 
her  hy  the  cube  of  the  denominator  of  the  fraction  which 
determines  the  degree  of  approximation;  extract  the  cube  root 
of  this  product  to  the  nearest  wiit,  and  divide  the  result  by 
the  denominator  of  the  fraction. 

2.  Find  the  cube  root  of     5  to  within  I.  .     .  Ans.  1|. 

3.  Find  the  cube  root  of  10  to  within  ?.  .     .  Ans.  2|. 

Since  the  cube  of  10  is  1000,  the  cube  of  100, 1000000, 
and. so  on,  the  number  of  ciphers  in  the  cube  of  the  denom- 
inator of  a  decimal  fraction  is  equal  to  three  times  the 
number  in  the  denominator  itself     Therefore, 

When  the  fraction  which  determines  the  degree  of  approxi- 
mation is  a  decimal,  add  tfiree  ciphers  for  each  decimal  place 
required;  and  after  extracting  the  root,  point  off  from  the 
right  one  place  of  decimals  for  each  three  ciphers  added. 

4.  Find  the  cube  root  of  2  to  five  places.     A.  1.25992. 

5.  Find  the  cube  root  of  3Y  to  six  places.  A.  3.332222. 

By  adding  ciphers  to  both  terms,  any  decimal,  or  whole 
number  and   decimal,   may   be  written   in   the   form    of  a 


154  RAYS  ALGEBRA,  SECOND  BOOK. 

fraction,  having  its  denominator  a  perfect  cube ;  thus, 
•2-7^'  -25-1^0,  6.4=f-Jgg,  and  so  on.  Therefore, 
to  find  the  cube  root, 

Annex  ciphers  to  the  given  decimal,  until  the  number  of 
decimal  places  shall  he  equal  to  three  times  the  number  re- 
quired in  the  root.  Extract  the  root,  and  point  off  from  the 
right  the  required  number  of  decimal  places. 

6.  Find  the  cub©  root  of  .4  to  four  places.     Ans.  .73G8. 

7.  Find  the  cube  root  of  84.3  to  six  places. 

Ans.  3.249112. 

To  find  the  cube  root  of  a  fraction  or  a  mixed  number, 
first  reduce  the  fraction  to  a  decimal. 

8.  Find  the  cube  root  of  |.    .     .     .    Ans.  .82207+. 

9.  Find  the  cube  root  of  5jg|.  .     .      Ans.  1.816+. 
10.  Divide  the  cube  root  of  ?AIli±2j  by  the  square  root 

32768  ''  ^ 

of  the  square  root  of  8.3521.  Ans.  -25. 


EXTRACTION    OF    THE    CUBE    ROOT    OF    ALGE- 
BRAIC   QUANTITIES. 

EXTRACTION    OF    THE  CUBE    ROOT    OF    MONOMIALS. 

lOO.  If  we  cube,  for  example,  2ax^,  we  have  (2rtf.T-)' 
=8a^c^ ;  that  is,  we  cube  the  coefiicient,  and  multiply  th« 
exponent  of  each  letter  by  3.  Hence,  conversely,  we  have 
the  followinoj 

Rule  for  Extracting  the  Cube  Root  of  a  Monomial.— 
Extract  the  cube  root  of  the  coefficient,  and  divide  the  expo- 
nent of  each  letter  by  3. 

Find  the  cube  root  of  the  followinir  Monomials : 


1.  ^x^z\     .     .    Ans.  2xz\ 

2.  21xY\ .     .  Ans.  3xy. 


3.  — 64rt'm«.      Ans.  —  4rtm^ 

4.  a^'^+'^x^  Ans.  a"'+V, 


EXTRACTION  OF  THE  CUBE  ROOT.        I55 

la  \3     a^  ./o^"     a     , 

Since,   ^-^^  =-p;    therefore,  ^-p  =  -5-     Hence, 

To  find  the  cube  root  of  a  monomial  fraction^  extract  the 
cube  root  of  both  terms. 

5.  Find  the  cube  root  of  '^^  ^ Ans.   0-2. 

Ztx^  6x 

6.  Find  the  cu?je  root  of  —  TrrF-^\r-i,-    •    Ans.  — -= — ^,. 


EXTRACTION  OF  THE  CUBE  ROOT  OF  POLYNOMIALS. 

lOl.  To  investigate  a  rule  for  extracting  the  cube  root 
of  polynomials,  let  us  first  examine  the  relation  that  exists 
between  a  polynomial  and  its  cube. 

(a_^6+c)3  =  { (a -f  6)  +  c}3  ^  (a  +  bf  +  {3(a  +  by-  +  3(a+6)c 

+  C2|C. 

(a+6-fc-fd)3={(a+6-^c)-fc^}3:=r(a+6+c)•'5+ 
|3(a+6-fc)24-3(a-f64-c)d-fd-}cZ. 

Hence,  the  cube  of  a  polynomial  is  formed  according  to  the  fol- 
lowing law : 

The  cube  of  a  j)olynomial  is  equal  to  the  cube  of  the  first 
term  —  plus  three  times  the  square  of  the  first  term,  plus  three 
times  the  product  of  the  first  term  by  the  second,  plus  the 
square  of  the  second,  all  'hree  midttplied  by  the  second  —  plus 
three  times  the  square  of  the  first  txco  terms,  plus  three  times 
the  product  of  the  first  two  terms  by  the  third,  plus  the  square 
of  the  third,  all  three  midtiplied  by  the  third,  and  so  on. 

By  reversing  this  law,  we  derive  the  following 

Rule  for  Extracting  the  Cube  Root  of  a  Polynomial.— 

1st.  Arrange  the  polynomial  with  reference  to  a  certain  letter. 

2d.  Extract  the  cube  root  of  the  first  term  for  the  first  term 
of  the  root,  and  subtract  its  cube  from  the  given  polynomial. 


156  RAY'S  ALGEBRA,  SECOND  BOOK. 

3d.  Take  three  times  the  square  of  the  first  term  of  the  root^ 
and  call  it  a  trial  divisor  for  finding  each  of  the  remain  in g 
terms  of  the  root.  Find  how  often  the  trial  divisor  is  con- 
tained in  the  first  term  of  the  remainder ;  this  will  give  the 
second  term  of  the  root.  Then  form  a  complete  divisor  hij 
adding  together  three  times  the  square  of  the  first  term  of  the 
root^  plus  three  times  the  product  of  the  first  term  hy  the  sec- 
ond^ plus  the  square  of  the  second.  Multiply  these  hy  the 
second  term  of  the  root^  and  subtract  the  product  from  the 
first  remainder. 

4th.  Again.,  find  how  often  the  trial  divisor  is  contained  in 
the  first  term  of  the  remainder ;  this  will  give  the  third  term, 
of  the  root.  Tlien  form  a  complete  divisor  as  before^  hy  add- 
ing together  three  times  the  square  of  the  first  and  second 
terms,  plus  three  times  the  product  of  the  first  and  second 
terms  hy  the  third,  plus  the  square  of  the  third.  Multiply 
these  hy  the  third  term  of  the  root,  and  subtract  the  product 
from  the  last  remainder. 

5th.  Continue  thus  till  all  the  terms  of  the  root  are  found. 

1.    Find    the   cube   root    of   x^—6a^-\-12x*-{-Sa'x*—Sx^ 
12a'x^^l2d'x'-\-Sa*x'—6a'x-{-a\ 

x<;_Gx^^l2x*-\-Sa^x^—8x^—12ci-x^~\-12a^x^-\-Sa^x^—Ga^x^a^ 
x'''  \x^—2x-\-a'^ 

Sx*—6x^  -f  4x^ )  —Gx^-j- 1 2a;4_8^ 
^—6x^+120:4— 8a;3 

Sx*—12x^'^12x'^-\-3a^x^—Ga^x^a^)-{^'Sa-x^—12a^x^-\-12a-x^ 
'  I  4-Sa^x^—6a^x4-a^. 

To  Lring  the  work  within  the  page,  the  last  n    •>    i      in    •?   i      1099 

remaiudrr  and  subtrahend  are  each  written  '    -T"^  "^       1-tt  X'  ~\-l^CC-X 
In  two  lines.  |  -j-3a^X^—Ga*X-\^Cl*'\ 


We  first  extract  the  cuhe  root  of  x^,  which  gives  x^  for  the  first 
term  of  the  required  root.  Then,  3  times  the  square  of  this,  =ioX*, 
constitutes  the  trial  divisor  for  finding  the  remaining  terms. 

Dividing  — Gx^  by  Sz"*,  gives  — 2x,  the  second  term  of  the  root. 
We  then  form  the  complete  divisor  by  adding  together  S[x-)^-]-S 
^x2x— 2a:)  +  (— 2a:)2^3a;4— 6a;3-f-4a;2.    Multiplying  this  by  —2x, 


EXTRACTION  OF  THE  CUBE  HOOT.        157 

and  subtracting,  the  first  term  of  the  second  remainder  is  -fSa^a;'*, 
which  divided  by  the  trial  divisor,  gives  -j-a^,  for  the  third  term 
of  the  root,  and  so  on.  ~ 


SECOND    METHOD. 

The  following  rule,  applicable  both  to  numerical  and 
algebraic  quantities,  may  be  found  more  convenient  in 
some  cases.  The  principle  upon  which  it  is  founded  will 
be  obvious  upon  a  careful  inspection  of  the  full  expansion 
of  the  forms  (a-\-by,   (c?-f  Z>-|-f)^,  etc. 

1.  Arrange  the  poli/nomial,  as  in  the  previoiis  rule. 

2.  Extract  the  cube  root  of  the  first  term,  etc.,  as  before. 

3.  Eind  the  trial  divisor  and  2d  term  of  the  root,  as  before. 

4.  Cube  the  root  already  found,  and  subtract  the  result 
from  the  given  polynomial. 

5.  Divide  the  first  term  of  the  remainder  by  the  same  trial 
divisor  for  the  third  term  of  the  root.  Cube  the  root  already 
found,  and  subtract  the  result  from  the  given  polynomial. 
Continue  this  process  until  a  quantity  is  found  in  the  root 
which  will  be  equal,  when  cubed,  to  the  given  polynomial. 

To  illustrate  this  rule,  take  the  example  given  above. 

icfi — 6a:'''-f  1 2a:4 -f-Sa^x^— 8a;-— 1 2a-a:H  1 2a2a;2-l- 3a%2_  ea^x-f- a« 

a;"*'  |2;2_2a:+a2 


^x^\—{ix-'-\-Vlx^^  etc.,  1st  remainder. 

a;fi  —Qx^-i^\2x^-^x\  cube  of  x^—2x. 

Si^l     Za^x^—Vla^x^,  etc.,  2d  remainder. 


a;'-._Ga:H12a;-»+3a2a:4_8a;-— 12a2.r-4-12a2a;2-f3a4a:2_6a^a:-|-a6 

We  first  extract  the  cube  root  of  X^,  and  find  it  x"^.  Cubing  this, 
subtracting,  and  dividing  the  first  term  of  the  remainder  by  SiC*,  we 
obtain  — 2x  for  the  second  term  of  tlie  root.  Cubing  x"^ — 2a:,  writ- 
ing it  below,  and  subtracting,  we  have  the  second  remainder. 
Dividing  the  first  term  of  this  remainder  again  by  SrC*.  we  obtain 
a^  for  the  third  term  of  the  root.  The  cube  of  a;2— 2a:-f-a2  being 
equal  to  the  given  polynomial,  the  work  is  fin'shed. 


158  RAYS  ALGEBRA,  SECOND  BOOK. 

Remarks. — 1.  A  second  method  for  extracting  the  square  root, 
similar  to  the  above,  might  be  given,  but  it  is  less  simple  than  the 
common  rule. 

2.  The  process  of  cubing  the  root  may  be  conducted  by  Newton's 
Theorem,  as  explained  in  Art.  172. 

Find  the  cube  root 

2.  Of  a^-f  24a^6-j-192«Z>2-}-5126'.  Ans.  a-l  86. 

3.  Of  Sa^—S4a'x-\-294ax'—S4Sx\         Ans.  2a— 1x. 

4.  Of  ««— 6c/5+15a*— 20a3+15a^— 6a-|-l. 

Ans.  a'—2a-\-l. 

5.  Of  x^—9a^-\-S9x'—d9x'-]-lb6x'—lUx-\-64:, 

Ans.  x'^ — 3.x-]-4. 

6.  Of  (a-]-ly"x'  —  6caP(a-\-iy'^x'-\-12chi'P(aJr'^y'' 
a;— 8c'a3^.  Ans.  (a-^iy"x—2caP. 

7.  Find  the  first  three  terms  of  the  cube  root  of  1 — x. 

.  ^       X      x^ 

Ans.  1 — ^ —  „ — 5  ^^^• 


IV.  EXTRACTION  OF  THE  FOURTH  ROOT,  SIXTH 
ROOT,  N'^"  ROOT,  Etc. 

]9!3.  The  fourth  root  of  a  number  is  one  of  four  equnl 
factors,  into  which  the  number  may  be  resolved  ;  and,  in 
general,  the  u'''  root  of  a  number  is  one  of  the  n  equal  fac- 
tors into  which  the  number  may  be  resolved. 

AVhen  the  degree  of  the  root  to  be  extracted  is  a  mul- 
tiple of  two  or  more  numbers,  as  4,  6,  etc.,  the  root  can  he 
obtained  hy  extracting  the  roots  of  more  simple  degrees. 

To  explain  this,  we  remark  that  {a^y=a^^^'—a^^,  and  in  gen- 
eral («*")"=.  a'"X«^a"i«.     Hence, 

The  TOL^^  iwwer  of  the  m'^  iioioer  of  a  number  is  equal  to 
the  mn'^  iiower  of  the  number. 

Reciprocally,  the  mn'^  root  of  a  number^  is  equal  to  the 
n'*  root  of  the  m'^'  root  of  that  number;  that  is, 


y  a—y,/  yd' 


EXTRACTION  OF  THE  CUBE  ROOT.       159 


From  this,  it  follows  that  ■^a=z:'\  y^a;    and    f/a='\  ;/«,  or 
\f^Ct;   in  like  manner  y'a=\-|/  ^/a,  and  so  on. 


1.  Find  the  4th  root  of  65536, 


Ans.  16. 


2.  Find  the  4th  root  of  ISlOv  .9601.    .     .  Ans.  10.7. 

3.  Find  the  6th  root  of  2985984.  .      .     .     Ans.  12. 

4.  Find  the  8th  root  of  214358881.     .     .     Ans.  11. 

5.  Find  the  4th  root  of  Sla*x^ Ans.  Sax\ 

6.  Find  the  4th  root  of  a*  -j-  4a'hx  -f-  Qa'b'x'  -\-  4:ah'x^ 
-\-h*x*.  Ans.  a-\-hx. 

1.  Find   the   4th   root   of  x^— 4x^-{-10x'  —  16x'-{-19 

x'^  x'^       x'^x^  ^x' 

8.  Findthe6thro«tofa«+-— 6/a*+-  1  +  15/  a^-f-  I 

-20.  ,  1 

Ans.  a . 

a 

193.  It  ha?  been  shown  already  (Arts.  182,  183)  that 
the  square  root  of  a  monomial,  or  a  polynomial,  may  he 
preceded  either  by  the  sign  -|-  or  — .  We  shall  now  ex- 
plain the  law  in  regard  to  the  roots  generally. 

If  we  take  the  successive  powers  of  -\-a  and  ~a,  we  have 

-fa,        -fa2^        -^a%        -fa''. 

—a,        4-a2_        _a3,        .^a\   .    .    .    -j  a-'",        ~a^"+K 

From  this  we  see  that  every  even  power  is  positive,  and 
that  an  odd  power  has  the  same  sign  as  the  root. 
Conversely,  it  is  evident, 

1st.  That  every  odd  root  of  a  monomial  miist  have  the  same 
sign  as  the  monomial  itself. 


Thus,  f -f8a»=-f2a,  f-8a^=—2a,  ■^— 32a»o=— 2a2. 

2d.   That  an  even  root  of  a   jmsitive  monomial  may  he 
either  jyosifive  or  negative. 


Thus,  ^'8la*b^^±:Sab\    f  G4ai2^dr2a^ 


16i7  RAYS  ALGEBRA,  SECOND  BOOK. 

3d.  That  every  even  root  of  a  negative  moiioTnial  is  im- 
possible; since  no  quantity  raised  to  a  power  of  an  even 
degree  can  give  a  negative  result. 

Thus,  y/ — a^,  ^  —  6,  {/  — c,  are  symbols  of  operations  which 
can  not  be  performed.  They  are  imaginary  expressions,  like 
l/^    ^116,  (Art.  182.) 


TO  EXTRACT  THE  N'^"  ROOT  OF  ANY  QUANTITY. 

104.  In  raising  any  monomial  to  the  ri'''  power,  (Art. 
172,)   we  raise   the  numeral   coefficient  to  the  n^^  power, 
and  multiply  each  exponent  by  ?i,  thus,  (2a^^*)^=8a^Z>^^. 
■  Hence,  conversely,  to  find  the  ?i'*  root  of  a  monomial, 

Extract  the  n<^  root  of  the  coefficient^  and  divide  the  expo- 
nent of  each  letter  hy  n. 

Rules  for  the  extraction  of  any  root  of  a  numerical  quantity,  or 
algebraic  polynomial,  may  be  formed  on  the  same  principle  as  is 
that  of  the  cube  root,  (Art.  191.)     Thus,  since 

(a+6)4z=:a4+4a''^^+6a262_|.4a63_^64^o^4^(4«3^.6a26+4a62 

-f63)6. 

(a_|_5)r.^a5_|_(5cj4_^10a36+10a252^5ci63_^^fN5^  etc. 

The  trial  divisor  for  the  fourth  root  would  be  of  the  form  4a-^,  or 
four  times  the  third  power  of  the  first  term  of  the  root,  and  the 
complete  divisor  of  the  form,  4a-'-f  6a26-^4a62_j_53_ 

For  the  fifth  root,  the  trial  and  complete  divisors  would  be  of  the 
forms,  5a^  and  l^a^-\\^o?'b-\-\^0?b-'\-^ab'^A^b''^  and  so  for  any 
higher  root. 

A  more  simple  method,  however,  would  be  like  that  which  is 
called  the  Second  Method  for  extracting  the  cube  root,  (Art.  191.) 
The  trial  divisors  would  be  of  the  form  4a^,  for  the  4th  root,  5a*  for 
the  5th  root,  na"-^  for  the  nth  root,  or,  in  general,  n  times  the  (n— 1)'^ 
power  of  the  first  term  of  the  root. 

Remark.— In  the  following  examples,  find  the  root  of  the 
numeral  coefficient  by  inspection.  It  is  Tinnecessary  to  give  rules 
for  extracting  the  5th,  7th,  etc.,  roots  of  numbers,  as  in  the  present 
state  of  science  these  operations  are  readily  performed  by  Loga- 
rithms. 


RADICAL  QUANTITIES 


1.  Find  the  5th  root  of  — 32( 


The  6th  root  of  7296V«. 
The  7th  root  of  128xy^ 
The  8th  root  of  Q'o6la'L'\ 
The  9th  root  of  — 512a;V^ 
The  10th  root  of  1024//°^= 


2. 
3. 
4. 
5. 
6. 
7.  Extract 


'b\ 


8.  Extract   the    5th    root    of    32x5- 
lOx— 1. 


-80, 


161 

Ads.  — 2ax-. 

Ans.  ±3Z>c^ 
.  Ans.  2x1/'^. 
Ans.  ±3aR 

Ans.  — 2xz'^. 

Ans.  ±26;.-\ 

.    Ans.  a'^bc\ 

;c*-f80x3— 40x'^ 
Ans.  2x — 1. 


V.    RADICAL    QUANTITIES. 

Note. — These  quantities  are  generally  called  surds  by  English 
writers;  while  the  French  more  properly  term  them  radicals,  from 
the  Latin  word  radix,  a  root. 

195.  A  Rational  Quantity  is  either  not  aiFected  by 
the  radical  sign,  or  the  root  indicated  can  be  exactly  as- 
certained ;  thus,  2,  a,  |/4,  and  |^8  are  rational  quan- 
tities. 

A  Radical  Quantity  is  one  affected  by  a  radical  sign, 
but  whose  indicated  root  can  not  be  exactly  expressed  in 
numbers;  thus,  |/5=2. 23606797  nearly. 

106.  From  Art.  193  it  is  evident  that  when  a  mono; 
mial  is  a  perfect  power  of  the  ?i"^  degree,  its  numeral  coeffi- 
cient is  a  perfect  power  of  that  degree,  and  the  exponent 
of  each  letter  is  divisible  by  n. 

Thus,  4a~  is  a  perfect  square,  while  Ga-^  is  not;  and  Sa*"'  is  a  per- 
fect cube,  while  6a-\  8a^,  la^\  etc.,  are  not. 

In  extracting  any  root,  when  the  exact  division  of  the  exponent 

can  not  be  performed,  it  may  be  indicated  by  a  fraction.     Ihus,, 

3  __  4 

■j/a*^  may  be  written  a^,  and  fa^  may  be  written  a^  ;  and,  in  gen- 

m 

eral,  the  n<A  root  of  the  m<'*  power  of  a  is  either  {/a"',  or  a ' . 
2d  Bk.  14 


162  RAY'S  ALGEBRA,  SECOND  BOOK. 

Since  a  is  the  same  as  a\   (Art  19,)  the  square  root  of  a  may 

be  expressed   thus,   a   ;    the  cube  root  thus,  a   ;  and  the  n<'*  root 

1 
thus,  a".     Hence,  the  following  expressions  are  equivalent: 


l/a  and  a^, 


_  1 

^a  and  a**. 


% 

Also,    ^a'^  and  a^, 

m 

y'a"^  and  «". 
Hence, 


77ie  numerator  of  the  fractional  exponent  denotes  the  power 
of  the  quantity^  and  the  denominator  the  root  to  he  extracted. 

lOT.  Theorem. — Any  quantity  affected  with  a  fractional 
exponent^  may  he  transferred  from  one  term  of  a  fraction  to 
the  other,  if  at  the  same  time,  the  sign  of  its  exponent  he 
changed. 

This  proposition  has  already  been  established  (Art.  81)  when  the 
exponent  is  integral.  It  is  also  true  when  the  exponent  is  frac- 
tional, as  we  shall  now  prove. 

Let  it  be  required  to  extract  the  cube  root  of  — . 

a^ 

As  \=a-'^  (Art.  81);    therefore,  ^/_=^?F2. 

But,  \/-l  =  i-  and  fa=^=a~i.     (Arts.  190,  194.) 
'  ^^     at 

1  2 

Therefore,  —-  =  «-#. 

af  \  m 

In  like  manner,  generally,  — 57j=:a— «  . 

198.  The  Coefficient  of  the  radical  is  the  quantity 
which  stands  before  the  radical  sign. 

Thus,  in  the  expressions  a|/6,  and  2]^c,  the  quantities  a  and  2 
are  called  coefficients. 

Radicals  are  said  to  be  of  the  same  degree  when  they 
have  the- same  index  ;  as,  a^  and  5^,  or  -^d'^  and  f^b'K 


RADICALS.  163 

Similar  radicals  have  the  same  index,  and  the  same 
quantity  under  the  radical  sign ;  as,  ay  b  and  cy^b ; 
Sfd'  and  5if ^. 

Before  entering  into  a  discussion  of  the  general  subject 
of  radicals,  it  is  important  to  observe  that, 

A  radical  quantity  is  raised  to  a  power  equal  to  the  index 
of  its  root,  by  simply  rejecting  the  radical  sign  with  its  index. 

Thus,  the  square  of  -/a  is  a,  the  cube  of  fa  is  a,  the  square 
of  y^3  is  3,  the  nth  power  of  y'a  is  a,  etc.  In  other  words, 
|/aXv/^=^>  fci'X^a'XfTi—a,  etc.  This  is  evident  from  the 
definition  of  a  root,  (Art.  173.) 


REDUCTION    OF    RADICALS. 

Case  I.' — To  beduce  Radicals  to  their  Simplest  Form. 

lOO.  Reduction  of  radicals  consists  in  changing  the 
form  of  the  quantities  without  altering  their  value.  It  is 
founded  on  the  following  principle  : 

The  square  root  of  the  j)i'oduct  of  two  or  more  factors  is 
equal  to  the  product  of  the  square  roots  of  those  factors. 

That  is,   |/a6=|/aXi/^;   which  is  thus  proved; 
Squaring  both  members  of  this  equation,  we  have,  (Art.  198,) 
ab=ayh,  or  ab—ab. 

Now,  since  the  equation  is  true  after  both  sides  are  squared,  it 
was  true  before,  (Art.  148,  Ax.  6,)  or  -^ab^yay^yb. 

By  this  principle,  ^/36=y'4x9=2x3 ;  v/144=ry'9><16=3x4; 
^8=|/4x2==-/4Xv' 2=2/2.     Hence,  we  have  the  following 

Eule  for  the  Reduction  of  a  Radical  of  the  Second 
Degree  to  its  Simplest  Form. — 1st,  Separate  the  quantity 
to  be  reduced  into  two  parts,  one  of  which  shall  contain  all 
the  factors  that  are  perfect  squares^  and  the  other  the  remain- 
ing factors. 


164  RAYS  ALGEBRA,  SECOND  BOOK. 

2d.  Extract  the  square  7'oot  of  the  perfect  square^  and  pre- 
fix it  as  a  coefficient  to  the  other  part  placed  under  the 
radical  sign. 

To  determine  whether  any  numeral  contains  a  fiictor 
fltat  is  a  perfect  square,  divide  it  by  either  of  the  squares 
4,  9,  16,  etc. 

Reduce  to  their  simplest  forms  the  radicals  in  each  of 
th«  following  examples : 


1.  ^12,    ^18,    ^45,    /32,   v^50^,    y72a2b\ 

Ans.  2/3,  3/2,  3/5,  4/2,  5a/2«,  6a6/25. 

2.  /245,    /i48,    /8T0,    /507Pc2,    /ISOSo^P. 

Ans.  7/5,  8/7,  9/10,  136c/36,  19a26/5. 

In  a  similar  mfinner,  polj'nomials  may  sometimes  "be  simplified. 


Thus,  /(3a3-6a2c+3ac2)=:/3a(a2— 2ac+c2)=r(a_c)/;ia. 


3.  /(as-a26),    ^ ax^—Gax-^9a,    ^ {x^--y2)i^r^y), 

Ans.  a/(a— 6),    (a:— 3) /a,    {x-^y)-^ {x—y). 

To  reduce  a  fractional  radical  to  its  simplest  form, 

1st.  Render  the  denominator  of  the  fraction  a  perfect 
square  hy  midtiplying  or  dividing  both  terms  hy  the  same 
quantity. 

2d.  Separate  into  two  factors^  one  of  which  is  a  perfect 
square. 

3d,  Extract  the  square  root  of  this  factor^  and  write  it  as 
a  coefficient  to  the  other  factor  placed  under  the  radical  sign. 

4.  Reduce  -|/|,  and  ^/  ,  to  their  simplest  forms. 
Vr-=»/|xl=>/l=/5V<5=v'i\X»/5--|/5. 

Ans.  ly/2,   J/c;   1/^   v/S,  3/30,  |/10. 


A6'      \56'      \4cV      \  9«sW 


RADICALS.  165 


^^^•Z^V^*^'    66^^^"^'    2cj,>^«2/,    14^2(6^^)   • 

200.  To  reduce  radicals  of  any  degree  to  the  most 
simple  form. 

The  principle  of  Art.  199  is,  evidently,  applicable  to 
radicals  of  any  degree.      Thus, 

1.  Reduce  -^54  to  its  most  simple  form. 


f54^f27x^=#27xf2=3^2. 


Similarly,   f  f-^iX|X|=f  i|=f  sVXlS-^f  18- 
Reduce  each  of  the  following  to  its  simplest  form : 


2.  f40,    f81c\    f  128afic5,    fie2m^n%    ^144. 

Ans.  2f  5,  3cf3c,   4a2cf  2c2,  Smnf6mn\  2^9. 


3-  fh    fl    fh    fi    V  f'    vi    vh  ^  _ 

Ans.  Ifi,    lf6,    ^f.36,    if  15,    1  f  5i,    |  V^,    Jf^32. 


4.  f  162,    f  3888,    ^Z'la^b^,    {/729a\ 

Ans.  3f'2,   6^3,   2a6jj'2a6'i,   3a (/3a. 

SOI.  The  mji^^  root  of  any  quantity  may  be  simplified 
when  it  is  a  complete  power  of  the  m^^  or  ji^^  degree,  as 
shown,  (Art.  192.) 


Thus,  l/9a^--^-\  |/9a2=/3a. 


Also,   l/a^-~2ab^b2=:\  ^a^~2ab-\~b^=fcr^. 
Reduce  each  of  the  following  to  its  simplest  form 


1.  ^36a2c2,   ^.Slin^n^    ^4a2,    f/16a2c4,    ^1256^. 

Ans.  |/6ac,   Sriy/lm,',   f2d,    f  4ac2,   ^65. 


106  RAY'S  ALGEBRA,  SECOND  BOOK. 

Case  II. — To  reduce  a  Rational  Quantity  to  the 
FORM  of  a  Radical. 

30S.  If  we  square  «,  and  then  extract  the  square  root 
of  the  square,  the  result  is  evidently  a. 

2  3 

That  is,  a==|/a2-__o^5      jn  ^^^  manner,  a=f^a^z=a^^  and  gen- 

m, 
erallj,    a='^ar=ar\     Hence, 

Rule  for  reducing  a  Rational  Quantity  to  the  form  of 
a  Radical. — Raise  the  quantity  to  a  jpower  corresponding  to 
the  given  root,  and  write  it  under  the  radical  sign. 

1.  Reduce  6  to  the  form  of  the  square  root.  Ans.  -j/BG. 

2.  — 2  to  the  form  of  the  cube  root.       Ans.  #^ — 8. 


3.  Sax  to  the  form  of  the  square  root.   Ans.  y^9aV^ 

4.  m — n  to  the  form  of  the  square  root. 

Ans.  -j/m"^ — Imn-^n^, 

Similarly,  a  coefficient  may  be  passed  under  the  radical  sign. 
Thus  2v/3=^4Xi/3=v/i^. 
Generally,  a'^b=^'l/O^X'V~b=V^^^- 

5.  Express  5]/T,  and  a^|///,  entirely  under  the  radical 
sign.  Ans.  |/iY5,  and  |/a*6. 

6.  Pass  the  coefficient  of  the  quantity  2|^5,  under  the 
radical  sign.  Ans.  ]^40. 

Case  III. — To  reduce  Radicals  having  Different 

Indices  to  Equivalent  Radicals  having  a 

Common  Index. 

203.  This  is  done  by  multiplying  both  terms  of  the 
fractional  exponent  by  the  same  number,  which,  evidently, 
does  not  change  its  value.     (Art.  118.) 


RADICALS.  167 

1 

Let, it  be  required  to  reduce  ^2a,  and  ]^36,  or  (2a) ^ 

and  (3^)'^   to  quantities  of  equal  value,  having  the  same 
index. 

f  36=  (36)4  =(36)  1^2:^12/ (36]3='^2763.     Hence, 

Hule. — Reduce  the  fractional  exponents  to  a  common  de- 
nominator; then  the  numerator  of  each  fraction  will  repre- 
sent the  power  to  which  the  corresponding  quantity  is  to  le 
raised,  and  the  common  denominator  the  index  of  the  root 
to  he  extracted. 


i  1.  Keducc  |/3  and  ^2,  or  3^  and  2^  to  a  common  index- 
Ans.  ^27  and  ;^  4,  or  27"  and  4". 

2.  -^5  and  |/4 Ans.  |"/25  and  y  64. 

J  

3.  a^  and  h" Ans.  |/a*  and  ^b. 

4.  r     if  a,   v^56,  and  ^6^. 

Ans.  1*/^%  \/62bh',  and  |>^2T6?. 

5.  r   y/^,  ^3/^,  and  if^.     Ans.  \^^«,  p^<  and  '^a\ 

6.  rteducc   3^,  2^,  and   5-  to  a  common  index. 

Ans.  3'^S  2^'^  5'",  or  \7B56i,  \^/512,  ^^15625. 

ADDITION  AND  SUBTRACTION  OF  RADICALS. 

!S04.  Required  to  find  the  sum  of  Sf/a  and  bf^a. 

It  is  evident  that  3  times  and  5  times  any  quantity,  must  make 
8  times  that  quantity;  therefore,  3f/a^5fa=8fa. 

But,  if  it  were  required  to  find  the  sum  of  S^/a  and  5f/fi,  since 
y  a  and  f^a  are  different  quantities,  we  can  only  indicate  their  addi- 
iion;  thus,  Sy/a^  bf/a. 


168  RAYS  ALGEBRA,  SECOND  BOOK. 

Similarly,  Sy2-^Ty2~4y2^Gy^. 
But  3/5  and  4j/a=3j/5-|-4|/3. 
So  also  3^/5  and  4^5^3/54-4^0. 

Radicals  that  are  not  similar,  may  often  be  made  so;  thus,  |/12 
and  |/27  are  equal  to  2^/S  and  3|/3,  and  their  sum  is  5|/3. 
The  same  principles  apply  to  the  subtraction  of  radicals. 
From  the  above  we  derive  the  following 

Rule  for  the  Addition  of  Eadieals.— 1st.  Reduce  the. 
radicals  to  their  simplest  forms,  and,  if  necessary,  to  a  com- 
man  index. 

2d.  If  the  radicals  are  similar,  find  the  sum  of  their 
coejicients,  and  prefix  it  to  the  common  radical;  hut  if  they 
are  not  similar,  connect  them  hy  their  proper  signs. 

Rule  for  the  Subtraction  of  Radicals. —  Change  the  sign 
of  the  subtrahend,  and  proceed  as  in  addition  of  radicals. 

1.  Find  the  sum  of  |/448  and  ^TVl. 


-/448=/64x7=  8/7 
|/T12=/T6X7=^V7 

By  addition,     12/7,  Ans. 

2.  Find  the  sum  of  f/^  and  f  81.      .     Ans.  5^3. 

3.  Of  ^48  and  f/102 Ans.  5^6. 

4.  Of  i/fS^i/^  and  yW^K  

Ans.   (3a-Z>-f-5aZ>)i/2aZ). 

5.  Subtract  i/TSO  from  |/405.  .     .     .     Ans.  3]/5. 

6.  Subtract  f/40  from  f/T3'5 Ans.  ^5". 

Perform  the  operations  indicated  in  each  of  tlie  follow- 
ing: 

7.  /243-f/27+^48. Ans.  16/3. 

8.  y^24-|-/54— /90. Ans.  /C 

9.  |/128— 2/50f/72-/18 Ans.  y'l 


RADICALS.  169 

10.  ^4Sab2-\-by''7Ea^i/Sa{a—9bj^ Ans.  a^Sa. 

11.  2„  f+^|/60+^T5+/|: Ans.   Vv/l^- 

12.  ^128— f686— f  16+4|f250 Ans.  15,^2: 

13.  2^14-81^2: Ans.  3f  2. 

14.  6f/4a2_^2|f2a+f/8a^. Ans.  9f  2ol 

15.  2/3— ^v/12+4y'27— 2^-3g Ans.  ^^^S. 

^U.  ^re+fST— 1^^=512+ f  192—7 f9 Ans.  10. 

17.  -x/^  +  i-/(«^^-4«^^^+4a63) Ans.  ^^ab. 


MULTIPLICATION  AND  DIVISION  OF  RADICALS. 

t20«S.  The  rule  for  the  multiplication  of  radicals  is 
founded  on  the  principle  (Art.  200)  that 

The  product  of  the  n*^  root  of  two  or  more  quantities  is 
equal  to  the  n"^  root  of  their  product. 

That  is,  VaxVb=Vcib.     (See  Art.  198.) 

Hence,  (Art.  53,)  a'i/bXov'd^aXcXv'^X\^d=acl/bd. 

The  rule  for  division  is  founded  on  the  principle  that 

The  quotient  of  the  n'^  roots  of  two  quantities  is  equal  to 
the  n'^  root  of  their  quotient. 

That  is,  J^r=  =  -\/'r;   which  is  thus  proved: 
'\/b       >^ 

Raising  both  sides  to  the  nth  power,  we  have  j-^^-ti  which  shows 
that  the  previous  equation  is  true.     Hence,  we  have  the  following 

Rules  for  the  Multiplication  and  Division  of  Radi- 
cals.— If  the  radicals  have  dijf'erent  indices,  reduce  them  to 
the  same  index.      Then., 

I.  To  l/Lulti-ply. —Multipli/  the  coefficients  together  for  the 
coefficient  of  the  product,  and  also  the  parts  under  the  radical 
for  the  radical  part  of  the  product. 
2d  Bk.  15* 


170  RAY  S  ALGEBRA,  SECOND  BOOK. 

II.  To  Divide. — Divide  the  coefficient  of  the  dividend  hy 
the  coefficient  of  the  divisor  for  the  coefficient  of  the  quotient^ 
and  the  radical  part  of  the  dividend  hy  the  radical  part  of 
the  divisor  for  the  radical  part  of  the  quotient. 

1.  Multiply  2i/«Z  by  Za^abc. 

2  ^ab 
Sa^abc 


2.  Divide  4ay'^ab  by  2|/c/c. 

4a£a6^  l?^«?=2a  J^=2avf =- ,/55. 
2i/ac'       2  ^  ac         \c         \c^       c  ^ 

3.  Multiply  2|f  3  by  Sy^. 

Multiplying,     .     .     .    6{/72,  Ans. 

4.  Divide  6^/2  by  3if  2. 

6/2=:6j/23^6^8.        (1.) 

3f2=:3^  22=3^4.         (2.) 

Dividing  (1)  by  (2),  wc  have  2f  2. 

5.  Multiply  3|/ 12  by  5^1^.      .     .     .     Ans.  90^^. 

6.  Multiply  4^/12  by  3|r4.    ....     Ans.  24^/6. 
T.  Multiply  together  5|/3,  7|/|^  and  |/2.  Ans  140 

8.  Multiply  Sfb  by  4 fa.      .     .     .       Ans.  12^/'^*^ 

9.  Multiply  together  y%  fS,  and  f  5.  A.  ^^648000. 

10.  Multiply  together  yx,  y^x\  and  i^'x^.     Ans.  yx', 

11.  Divide  ^40  by  i/2. Ans.  2fZ. 

12.  Divide  6^/54  by  3y^ Ans.  6y  3. 


RADICALS. 


171 


.  Ans.  5^4 
.  Ans.  f/S. 
.  Ans.  2f/  3. 

.    Ans.  |/2. 
b 


Ans 


■v.- 


13.  Divide  70f  9  by  7^l8 

14.  Divide  f/72  by  y2. . 

15.  Divide  4f9  by  2]/ 3. 

16.  Divide  f  72  by  1^3.. 

17.  Divide  ^[  by  ^^.    . 
Polynomials  containing  radicals  may  also  be  multiplied;  thus, 

18.  Multiply  3+V/5  ^y  2—1/5. 

2—    y'S 

6+2/5" 

6—  ^5— 5=1— 1/5,  Ans. 

19.  Multiply  y/2+1  by  |/2— 1 Ans.  1. 

20.  ll|/2— 4^15  by  y^  6-(-|/  5.  Ans.  2|/3— v^lO. 

21.  Raise  |/2-i-y'3  to  the  4th  power.  Ans.  49-|-20y  6. 


22.  Multiply  r/l2+|/19  by  l/l2— 1/19.         Ans.  5. 

23.  Multiply  a!'— .^1/2+1  by  x'-{-xy/2-{-l.  Ans.  a->-f  1. 

24.  (x^-f  l)(x2— ic^  3+l)(x^-fa:|/34-l).       Ans.  x«-f  1. 

SOO.  To  reduce  a  fraction  whose  denominator  contains 
radicals,  to  an  equivalent  fraction  having  a  rational  denom- 
inator. 

When  the  denominator  is  a  monomial,  as  ,  it  will  become  ra- 

tional  if  we  multiply  both  terms  by  y/O. 

Again,  if  the  denominator  is  fa,  if  we  multiply  both  terms  by 
f  a'^,  the  denominator  will  become  f  a'Xf/a'^^i^a^^za. 


Thus,    ^_ 


172 


RAY'S  ALGEBRA.  SECOND  BOOK. 


In  like  manner,  if  the  denominator  is  *\/ci^j  it  will  become  ra- 
tional by  multiplying  it  by  "y^a'"-".     Therefore, 


When  the  denominator  of  the  fraction  is  a  mo7iomial, 
7nultiply  both  terms  hy  such  a  factor  as  will  render  the  ex- 
jwnent  of  the  quantity  under  the  radical  equal  to  the  index 
of  the  radical. 

Since  the  sum  of  two  quantities,  multiplied  by  their  difference,  is 
equal  to  the  difference  of  their  squares  (Art.  80);  if  the  fraction  is 

of  the  form  , ,  and  we  multiply  both  terms  by  b — i/c,  the  de- 

nominator  will  be  rational. 


Thus, 


_        a{b—^c)       ^ah~a^G 
6+i/c~(6+|/c)  (&-- i/c^""     ly'—G 

If  the  denominator  is  6 — |/C,  the  multiplier  will  be  b-\-y'c.  If 
the  denominator  is  ^6-f-i/c,  the  multiplier  will  be  y^6 — |/c;  and 
if  it  is  |/6 — |/c,  the  multiplier  will  be  y^6+|/c. 

If  the  denominator  is  of  the  form  ^a-^-y/b-^-y'c,  it  may  be 
rendered  rational  by  two  successive  multiplications.  The  first  will 
result  in  a  quantity  of  the  form  m — ^/n,  which  may  be  made  ra- 
tional as  before. 


Reduce  the  following  fractions  to  equivalent  ones  hav- 
ing rational  denominators : 


i.-L 

1/3- 


fS' 


Ans.  ^-=Ji/3. 


2.  ^.  Ans.  yl^=W2: 

/6  ^ 


Ans.  I] 


,      6  __ 

4.  r-.  Ans.  11^16. 

5.  tV^.  Ans.  i-^y2. 
3-2/2 

6.  V^±V?.  Ans.  r)-U2./6. 
V/3-/2 


^   3/5-2/2 
'  2/5^/18' 
8         3-hy3"   ^ 

•  /6-f-/2-/5' 


.      Ans.  9+1/10. 
Ans.  -/G-f /2-f-/5. 


RADICALS.  173 


9.  ^=:z  4- - Ans.  2a:. 

x^^x^—\   '  x—^x^—\ 


10.  '^!!±'_+'^_"_!z^  +  i^"!±i-'^^ An.  2.^. 

^x^^\—^x^-\      yx^-j^l-{-^x^—l 


Remark. — By   the  preceding  transformations,   the  process  of 
finding  the  numerical  value  of  a  fractional  radical  is  very  much 

2 
abridged.     Thus,  to  find  the  value  of  ~~^,  we  may  divide  2  by  the 

square  root  of  5,  which  is  2.2360679+.     But  ~=  =  J^,  the   true 

yo         5 
value  of  which  is  found  by  multiplying  2.2360679  by  2,  and  divid- 
ing the  result  by  5. 

Reduce  each   of  the  following  fractions  to  its  simplest 
form,  and  find  the  numerical  value  of  the  result : 

11.  -^,  and  — ^ Ans.  .894427+,  and  .707106+. 

,„    v/25+,/12 


POWERS    OF    RADICALS. 
SOT.  Let  it  be  required  to  raise  i^Sa  to  the  3d  power. 
Taking  p'Sa  as  a  factor  three  times,  we  have 

So,  "^aX^VctX^cT  ...     to  n  factors,  =7a\     Hence, 

Hule  for  raising  a  Radical  Quantity  to  any  Power.— 

Raise  the  qiiantitij  under  the  radical  to  the  given  power ^  and 
affect  the  result  icith  the  primitive  radical  sign. 

If  the  quantity  have  a  coefficient,  it  must  also  be  raised  to  the 
given  power.  Thus,  the  4th  power  of  2f/M^Js_16f81a>^.  This, 
by  reduction,  becomes  16f  27a6x3a2^48a2f  3a2. 


174 


RAY'S  ALGEBRA,  SECOND  BOOK. 


If  the  index  of  the  radical  is  a  multiple  of  the  exponent  of  the 
power,  the  operation  may  be  simplified.     Thus, 


{p2ay^={J  y2aj^=^2:a,  (Art.  192.) 

In  general,  Cv'TTf  ^(  ^^a  )  ^'^'^^a.     Hence, 

If  the  index  of  the  radical  is  divisible  hy  the  exponent  of 
the  power,  we  may  perform  this  division,  and  leave  the  quan- 
tity under  the  radical  sign  unchanged. 

Thus,  to  raise  ^  3a  to  the  4th  power,  we  have  ^  81a^^  \    y 
r=:y^3a,  or,  dividing  8  by  4,  we  obtain  at  once  y'Sa. 


81  a4 


1.  Kaise  f^2a  to  the  4th  power. 

2.  S^2ab^  to  the  4th  power. 
■^ac^  to  the  2d  power. 
l/ac''  to  the  4th  power.    . 
■y/Sc'^  to  the  3d  power.     . 


yx — y  to  the  3d  power. 


.  .  Ans.  2af/2a. 
Ans.  162ahY^aI/. 
.  .  .  Ans.  cy'a. 
.  .  .  Ans.  aV. 
.     .     .     Ans.  cy'S. 


Ans.  (x — y)\/^ — y^ 


ROOTS    OF    RADICALS. 

208.   Since  !;!/  l/a^y'a  (Art.  192),  therefore,  to 
tract  the  roots  of  radicals,  we  have  the  following 


ex- 


Rule. — Multiply  the  index  of  the  radical  hy  the  index  of 
the  root  to  he  extracted,  and  leave  the  quantity  under  the  radi- 
cal sign  unchanged. 

Thus,  the  square  root  of  f^2d  is  ^   ^2a=^2a. 
If  the  radical  has  a  coefficient,  its  root  must  also  be  extracted. 
If  the  quantity  under  the  radical  is  a  perfect  power  of  the  same 
degree  as  the  root  to  be  extracted,  the  process  may  be  simplified. 


Thus,  ^  ^8^3  is  equal  (Art.  192)  to  -^  f'8a3-=^2a. 


RADICALS.  175 

1.  Extract  the  cube  root  of  y^d^b.  .     .  .     Ans.  \/d^h. 

2.  The  4th  root  of  16a«lf  2^.       .     .  Ans.  2^^^  ^  2^ 

3.  The  square  root  of  fWa}.       .     .  .      Ans.  f^Ta. 

4.  The  cube  root  of  64fSlF.       .     .  .  Ans.  4^^ 2^. 

5.  The  cube  root  of  (m-\-n')y'm-\-n.  Ans.  y^m-\-n. 


IMAGINARY,  OR  IMPOSSIBLE  QUANTITIES. 

SCO.  An  imaginary  quantity  (Arts.  182,  193)  is  an 
even  root  of  a  negative  quantity. 

Thus,  |/  — a,  and   y^— 6'',  are  imaginary  quantities. 
The  rules  for  the  multiplication  and  division  of  radicals  (Art.  205) 
require  some  modification  when  imaginary  quantities  are  to  be  mul- 
tiplied or  divided. 

Thus,  by  the  rule  (Art.  205),  |/"^aX  i/— «=>/— «X— «— 
ya^—ztia.  But,  since  the  square  root  of  any  quantity  multiplied 
by  the  square  root  itself,  must  give  the  original  quantity,  (Art.  198,) 
therefore,  y/^^XV~^=~^- 

SIO.  Evert/  imaginary  quantity  may  he  resolved  into  two 
factors,  one  a  real  quantity,  and  the  other  the  imaginary  ex- 
pression, |/ — 1,  or  an  exj)ression  containing  it. 

This  is  evident,  if  we  consider  that  every  negative  quantity  may 
be  regarded  as  the  product  of  two  factors,  one  of  which  is  — 1. 
Thus,  — a=raX  — 1,    -^^r^ft^^-l,  and  so  on. 


Hence,  ^— a2:=p/a^X— l=i/«^Xi/— l  =  =t=a/  — 1. 

Since  the  square  root  of  any  quantity,  multiplied  by  the  square 
root  itself,  must  give  the  original  quantity; 

Therefore,  (^:rTy2^  ^ZTXi/^^=— 1. 

Also,  (^3I]3_(^^2x^^=:=-Vlir=— i/^TT. 

(>/-i)M/-i)' (v/-i)M-i)(-i)=+i- 

Attention  to  this  principle  will  render  all  the  algebraic  opera- 
tions, with  imaginary  quantities,  easily  performed. 

Thus^  ^11^  X  /=&  ---/ax  V—^  X  i/6  X  i/^  =  >/«6x 


176  RAY'S  ALGEBRA,  SECOND  BOOK. 


OPERATION. 


If  it  J)e^  required  to  find  the  product  of  a+6^/— 1 

a-j-by/ — 1  by  a — 6/^,  the  operation  is  a—h^ — 1 

performed  as  in  the  margin.  a'^-\-ab  /~~1 


a^-^b\ 

Since  a^-\-b'^-=z[a^b^'—i){a — 6^/— 1),  any  binomial  whose 
terms  are  positive  may  be  resolved  into  two  factors,  one  of  whicli 
is  the  sum  and  the  other  the  difference  of  a  real  and  an  imaginary 
quantity. 

Thus,  m-Yn={ym-\-^n'^—l)[^m^^lfi^^^)^ 


1.  Multiply  -j/ — a^  by  -p,/ — h'^ Ans.  — ah. 

2.  Find  the  3d  and  4tli  powers  of  ay/^l. 

Ans.  — a^\/- — 1,  and  a*. 
8.  Multiply  2,/^  by  2.^'-^.  .  .  Ans.  — 6|/ 6. 
4.  Divide  6^/^=^  by  2|/=4.     ....  Ans.  ly'E. 

6.  Simplify  the  fraction  z "r^'       •     -^"^-  )/ — 1- 

6.  Find    the    continued    product   of   x-\-a,    x-\-ay — 1, 
X — «,  and  X — a-j/ — 1.  Ans.  cc* — a*. 

7.  Of  what  number  are  24+ Y|/^,  and  24—7]/^, 
the  imaginary  factors?  Ans.  625. 


VI.    THEORY    OF    FRACTIONAL    EXPONENTS. 

211.  The  rules  for  integral  exponents  in  multiplica- 
tion, division,  involution,  and  evolution,  (Arts.  56,  *70, 
172,  and  194,)  are  equally  applicable  when  the  exponents 
are  fractional. 

Fractional  exponents  have  their  origin  (Art.  19G)  in  the 


FRACTIONAL  EXPONENTS.  177 

extraction  of  roots,  when  the  exponent  of  the  power  is  not 
divisible  by  the  index  of  the  root. 

Thus,  the  cube  root  of  a-  is  a^.     So  the  n^'^  root  of  a"'  is  a". 

2         4  '" 

The  forms  a^,  a^,  and  a    '*,  may  be  read  a  to  the  power  of  f, 

771 

a  to  the  power  of  |,  and  a  to  the  power  of  minus  — ;  or,  a  expo- 
nent  |,  a  exponent  |,  a  exponent 


MULTIPLICATION  AND   DIVISION   OF    QUANTITIES  WITH 
FRACTIONAL  EXPONENTS. 

2X2»  It  has  been  shown  (Art.  56)  that  the  exponent 
of  any  letter  in  the  product  is  equal  to  the  sum  of  its  expo- 
nents in  the  two  factors.  It  will  now  be  shown  that  the 
same  rule  applies  when  the  exponents  are  fractional. 

2  4 

1.  Let  it  be  required  to  multiply  a^   by  a**. 

a^^f^^='^/~a'\   a^^l/a*=.'(^^,  (Art.  205.) 

2  4  33 

But  this  result  is  the  same  as  that  obtained  by  adding  the  expo- 
nents together. 

2         4         2.4         111+13         22 

Thus,  a3><a5^ct^    -"^^a'^    "S^ain. 
Hence,  where  the  exponents  of  a  quantity  are  fractional, 

To  Multiply,  Rule. — Add  the  exponents. 

_3  5 

2,  Let  it  be  required  to  multiply  a  *   by  a^. 

Adding  — |  and  |,  we  have  J,.    Hence,  the  product  is  a^,  or  ^^  a] 

SIS.  By  an  explanation  similar  to  that  given  in  the 
preceding  article,  we  derive  the  following  rule.  Where  the 
exponents  of  a  quantity  are  fractional, 

To  Divide,  Rule. — Subtract  the  exponent  of  the  divisor 
from  the  exponent  of  the  dividend. 


178  RAYS  ALGEBRA,  SECOND  BOOK. 

Perform  the  operations  indicated  in  each  of  the  follow- 
ing examples : 

1  2  _1  2  7  1 

1.  a2^a3,  and  a  ^ya^ Ans.  a^,  and  a^. 

3  3  U    -3 

2.  aAc-^yjo^c^ Ans.  a  4  c  ». 

2  112         11 

4.  (a3-f  a363-f  63)(a3_63) Ans.  a—b. 

5.  {x^y\y^)  {x^—y~3) Ans.  x^y—y^ 


inn 


6.  (a+6)"»X(«4-^)"X(«— ^)'"X(«— ^r-    Ans.  (a2— 62) 

2  1  1  2^  _5  in—vn 

7.  a;3_=_a:4^  and  x'^y'^^x^y^.  .    .    Ans.  rci^,  and  x  '""  i/"-"" 

8.  (a4— 64)^. (a?— 64) Ans.  a2_^a464+62 

9.  (a__62)^(a4^a26^-a^64-62.) Ans.  a?— 6^ 


POWERS  AND   ROOTS    OF   QUANTITIES  WITH  FRACTIONAL 
EXPONENTS. 

I214.  Since  the  m'''  power  of  a  quantity  is  the  product 

of  m  factors,  each  equal  to  the  quantity  (Art.  172); 

I 
Therefore,  to  raise  «»  to  the  mih  power,  we  have 

1-        L        L  VI 

a'*X"'*X^"   •     •     •     to  m  factors  =«'•. 

Hence,  to  raise  a  quantity  affected  with  a  fractional  ex- 
ponent to  any  power, 

E^llle. — Multiply  the  fractional  exponent  hy  the  exponent 
of  the  poiver. 

Thus,  (ai63)*=a263=a265. 

215.    Conversely,    to   extract   any   root   of  a  quantity 
affected  by  a  fractional  exponent, 


EQUATIONS  CONTAINING  RADICALS.  179 

Rule. — Divide  the  exponent  hy  the  index  of  the  root. 

_m  m  »"     1  1 

Thus,  y an=an-^'^^a^''^=w^. 


1.  Raise  crh^  to  the  4th  power.     .  ' .     .    Ans.  a^h'^. 


]    1    1 


2.  Raise  — ^xry^z'^  to  the  3d,  4th,  and  6th  powers. 

3       3  4  3 

Ans.  —^x'-yz"^  ;  l^xY^]  QixYz'^. 
1 

3.  Find  the  square  of  a — (aic — a^)^. 

1 
Ans.  ax — 2a(ax — a^)^. 

1  —J 

4.  Find  the  cube  of  a^x'^-\-a  '^x. 

Ans.  ax-^-\-Sa''^x-'^-\-Sa  '^x-\-a-^x^. 

5.  Find  the  cube   roots  of  (2la^xy  and  (2Y«'.t)^. 

Ans.  sUic^  or  (3aa;"3)i  ;  and  (3ax^)i 

1 

6.  Find  the  square  root  of  5x^ — 4a;(5caj)^-f-4c. 

Ans    sM— 2A 

7.  Find  the  cube  root  of  {^a^—^,a"b^-{-Gab—Sb^. 

Ans.  ^a— 26^. 

VII.    EQUATIONS    CONTAINING    RADICALS. 

SIG.  In  the  solution  of  questions  containing  radicals, 
the  method  to  be  pursued  will  often  depend  on  the  judg- 
ment of  the  pupil,  as  many  of  them  can  be  solved  in  dif- 
ferent ways,  and  the  shortest  processes  can  only  be  learned 
from  practice. 

1st.  When  the  equation  to  be  solved  contains  only  one 
radical  expression,  transpof=e  it  to  one  side  of  the  equation 
and  the  rational  terms  to  the  other;  then  involve  both 
sides  to  a  power  corresponding  to  the  radical  sign. 


180  RAY  S  ALGEBRA,  SECOND  BOOK. 

1.  Given,  ]^(a^-f^) — a=c,  to  fiud  x. 

Transposing,  ^(a^-)-a:)=c-|-a; 

Cubing,  a^-{'X=c^-\-Sac^-]-3a^c-{-a^', 

Whence,  x^c^-^^Sac^ySa^c. 

2d.  When  a  radical  expression  occurs  under  the  radical 
sign,  the  operation  of  involution  must  be  repeated. 


2.  Given 


X — 1/1 — x=:l — i/x,  to  find  X. 


Squaring,     x—^/l — x=l — 2i/x-\-X] 
Canceling  x  on  each  side,  and  squaring  again, 

l—x=l—4y'x'-]-ix. 

Canceling  1  on  each  side,  transposing,  squaring,  and  reducing, 

We  find,  x=U. 

8d.  When  there  are  two  or  more  radical  expressions,  it 
is  generally  preferable  to  make  one  of  them  stand  alone 
before  performing  the  process  of  involution. 


3.  Given,  |/.T-|-9 — y^x=l,  to  find  x. 

Transposing,  — |/a:,  we  have  |/ic-|  9=l  +  v^iC. 
Squaring  each  side,  x-\-9=z1-\-2y/x-\-x; 

Canceling  X  on  each  side,  transposing,  and  dividing  by  2, 
^x=A:;  hence,  a;™16. 

In  some  cases,  however,  it  is  preferable,  when  an  equation  con- 
tains two  radical  expressions,  to  retain  them  both  on  the  same  side. 
Thus,  the  following  equation  will  be  cleared  of  radicals  at  once,  by 
squaring  each  side : 

\  \  x—a  /    '     \  \  x-\--a  J  r/6-— 4 

4.  ■/(a:+5)-f3^8— /Z Ans.  a:=4. 


5.  Jl  +  v/(3+v/6a;)=2 Ans.  a:=G. 

6.  i/ic+a=/^-j-a Ans.  x=r   ~  '  . 

7.  /2x— 3a -f  /2x=3 /a Ans.  a;=2a. 


INEQUALITIES 


8.  /{13+v/[7+/(3+^^)]}:=4. 
_         4 


9.  y2-\-x^'^/x-. 


y2-j-a;' 


10.  y/a-{-x-^^j-^YX. 


11.  i/x-\-l'S—yx—n=2. 


12.  a  ]/x  f  b  y'  X — c  ^  x=d. 


13. 


x—ax     ^/x 


14.  a:+a=|/a^-t-a;/(62_i_a;2). 


3x-l 


3 

;/^-l 


^"-  ya:c+l-^+       2 


18.  |/4a-^a:=2|/6+a:— /a;. 


19    ^/    ^      I      /    c     ^4/  46c 


20.     ^^=rr-i-'^^c. 

yx^a—y/x 


,/a:-f  3— Y  ^/.t— 3=:^'  2  |  x. 

JL+1 


21.  V  1 

22    L.L_J{i  + J(A_,1U 
a:  ^a"  A    la2^  A  \  hix;^^  x^  //' 


Ans.  x^= 


181 

.  Ans.  x=\. 
.  Ans.  a:=^2. 
a 


Ans.  X 

.     ,  Ans.  a: 
Ans.  X 


a-^2i/  a 
Ans.  a;:=:;36. 

d^ 
(a4  6-c)2* 
1 
1-a 
62-4a2 


4a 


Ans.  a:=z:-l4i. 


Ans.  o'^lGa. 


.     .    Anp.  a:=:3. 


Ans.  X 
Ans.  a:- 

Ans.  X- 

Ans.  a:- 


2a— 6 

_a(6+c) 
~   b—G   ' 

_a(c-])2 

Ans.  a:=r9. 
Aab'^ 


a^—ib'^' 


23.  -,,/(l-|-a)2-]-(l— aja;-i-j,/(l— aj2_j-^^l-j-aja;_2a.  Ans.  a;--_8. 


VIII.    INEQUALITIES. 

SIT*  In  the  discussion  of  problems,  it  often  becomes 
necessary  to  compare  quantities  that  are  inieqnal^  and  to 
operate  upon  tbem  so  as  to  determine  the  values  of  the 
unknown  quantities,  or  to  establish  certain  relations  be- 
tween them. 


182  RAY'S  ALGEBRA,  SECOND  BOOK. 

In  most  cases  the  methods  of  operating  on  equations 
apply  to  inequalities,  but  there  are  some  exceptions. 

318.  In  the  theory  of  inequalities,  it  is  convenient  to 
consider  negative  quantities  less  than  zero. 

In  comparing  two  negative  quantities,  that  is  considered 
the  least  which  contains  the  greatest  number  of  units  ; 
thus,  0>— 1,  and  — 3>— 5. 

Two  inequalities  are  said  to  subsist  in  the  same  sense, 
when  the  greater  quantity  stands  on  the  right  in  both,  or 
on  the  left  in  both  ;  as,  5>3  and  7>>4. 

Two  inequalities  are  said  to  subsist  in  a  contrary  sense, 
when  the  greater  stands  on  the  right  in  one  and  on  the 
left  in  the  other ;  as,  5>1  and  4<8. 

SIO.  Proposition  I. — If  the  same  quantity^  or  equal 
quantities^  he  added  to  or  subtracted  from  both  members  of 
an  inequality,  the  resulting  inequality  will  continue  in  the 
same  sense. 

Thus, 7>5. 

Adding  4  to  each  member,      .     .     .  11>9. 
Subtracting  4  from  each  member,    .     3^1. 

Also,  — 5<^ — 3;  and  by  adding  and  subtracting  4, 

— l<-fl,  and  — 9<--7. 
Similarly,  if  a>6,  then  a-f  c>6-f-c,  or  a— c>6 — c.     Hence, 

Any  quantity  may  be  transposed  from  one  side  of  an  in- 
equality to  the  other,  if  at  the  same  time  its  sign  be  changed. 

S20.  Proposition  II. — If  two  inequalities  exist  in  the 
same  sense,  the  corresponding  members  may  be  added  together^ 
and  the  resulting  inequality  will  exist  in  the  same  sewse. 

Thus,  if  7>6,  and  5>4;  then, 
74-5>6+4,  or  12>10. 

When  two  inequalities  exist  in  the  same  sense,  if  we 
subtract    the    corresponding    members,    the    resulting    in- 


INEQUALITIES.  183 

equality  will  exist,  sometimes  in  the  same,  and  sometimes 
in  a  contrary  sense. 

First,  7>3  By  subtracting,  we  find  the  resulting  inequality 

4>-l         exists  in  the  same  sense. 

3>2 

Second,  10>9  In  this  case,  after  subtracting,  we  find  the 

8>3  resulting  inequality  exists  in  a  contrary 

~2<6  ^^''^^• 

In  general,  if  a>6  and  c>d,  then,  according  to  the  particular 
values  of  a,  6,  c,  and  d,  we  may  have  a—c^b—d,  a — c<6 — d, 
or  a~c=b — d. 

S31.  Proposition  III. — If  the  two  members  of  an  in- 
equality be  midtij)lied  or  divided  by  a  positive  number,  the 
residting  inequality  icill  exist  in  the  same  sense. 

Thus,  8>4  and  8X3>4x3,  or  24>12. 
Also,  8--2>4--2,  or  4>2. 

This  principle  enables  us  to  clear  an  inequality  of  frac- 
tions. 

If  the  multiplier  be  a  negative  number,  the  resulting 
inequality  will  exist  in  a  contrary  sense. 

Thus,  — 3<— 1,  but  _3X— 2>— lX-2,  or  6>2. 

From  this  principle  we  derive 

23S.  Proposition  IV. —  TJie  signs  of  all  the  terms  of  both 
members  of  an  inequality  may  be  changed,  if  at  the  same  time 
toe  establish  the  residting  inequality  in  a  contrary  sense. 

For  this  is  the  same  as  multiplying  both  members  by  — 1. 

SI33.  Proposition  V. — Both  members  of  a  positive  in- 
equality may  be  raised  to  the  same  power,  or  have  the  same 
root  e^Jracted,  and  the  resulting  inequality  icill  exist  in  the 
same  sense. 

Thus,  2<3  and  22<32,  2'^<<^^    or  4<'0,  8<27;  and  goon. 
Also,  25>16,  and  /25>i/16,  or  5>4;   and  so  on. 


184  RAY'S  ALGEBRA,  SECOND  BOOK. 

But  if  the  signs  of  both  members-  of  an  inequality  are 
not  positive,  the  resulting  inequality  may  exist  in  the  same, 
or  in  a  contrary  sense. 

Thus,  3>— 2,  and  32>(— 2)2,  or  9>4. 
But,  — 3<-2,  and  (_3)2>(— 2)2,  or  9>4. 


EXAMPLES  INVOLVING  THE  PRINCIPLES  OF  INEQUALITIES. 

1.  Five  times  a  certain  whole  number  increased  by  4,  is 
greater  than  twice  the  number  increased  by  19  ;  and  5  times 
the  number  diminished  by  4,  is  less  than  4  times  the  num- 
ber increased  by  4.     Required  the  number. 

Let  a:=  the  number. 
Then,  5a:+4>2a;+19,  (1) 

5a:— 4<4a:+4.  (2) 

5a:— 2a:>19— 4,  from  eq.  (1)  by  transposing, 
3a:>15,  by  reducing, 
3:^5,  by  dividing  both  members  by  3. 
5a:— 4a:<4+4,  from  eq.  (2)  by  transposing, 
^<8,  by  reducing. 

Hence,  the  number  is  greater  than  5  and  less  than  8,  consequently 
eithei-  6  or  7  will  fulfill  the  conditions. 

'2.   If  4x— 7<2.2:-f  3,   and   3a:+l>13— rr,  find  x. 

Ans.  a-=4. 

3.  Find  the  limit  of  x  in  7a:— 3>32.         Ans.  a:>5. 

4.  Of'a:  in  the  inequality  5-f-|a;<84-|a;     Ans.  a:<36. 

(Z—\—C  —\—6 

5.  Show  that     J~  ~|~   >  the  least,  and  <  the  greatest 

d      C       € 

of  the  fractions,  j,   ^,   -.,  each  letter  representing  a  posi- 
tive quantity. 

Suppose  y-  to  be  the  greatest,  and  -^  the  least,  of  the  fractions, 
ace  ace      c    e     c        ,a     a    c   ^a    e  ^a 

I'  d  J  ^"'="'  -b>a  a=rf  />rf'  ""*  6=6'  d<v  j<b- 


EQUATIONS.  185 

«>t'   "^tI''   ">!•     (^'•'•221-) 

ab         ad         af 
a  =  -^,  c<-^,   e<-^.     (Art.  221.) 

a  +  c  +  e>(6fc?4-/)^.     (Arts.  219,  220.) 

a_{_c  +  €<(64-d+/)^.     (Arts.  219,  220.) 

a_(_c  +  e     c  a  +  c-fe     a 

Hence,  r—, — ^rr-j^^  ^i  J    and  ^ *— r-:c<  t"- 

'  b-\-d^S     d'  b-\-d-{-f^b 

6.  It  is  required  to  prove  that  the  sum  of  the  squares 
of  any  two  unequal  magnitudes  is  always  greater  than  twice 
their  product. 

Since  the  square  of  every  quantity,  whether  positive  or  negative, 
is  positive,  it  follows  that 

(a_6)2,  or  a2— 2a6+62>0. 
Adding,  -\-2ab  to  each  side  (Art.  219), 

a2_|_62^2a6,  which  was  required  to  be  proved. 

Most  of  the  inequalities  usually  met  with,  are  made  to  depend 
ultimately  upon  this  principle. 

7.  Which  is  greater,  y'S+i/Ti  or  yS-^S^/T? 

Ans.  the  former. 

8.  Given  J(a;+2)-|-Jrc<J(a:— 4)-f 3  and  >-^(a:+l)-|-», 
to  find  X.  Ans.  x=:S. 

9.  The  double  of  a  certain  number  increased  by  7,  is 
not  greater  than  19,  and  its  triple  diminished  by  5,  is  not 
less  than  13.     Required  the  number.  Ans.  6. 

10.  Show  that  every  fraction  -f-  the  fraction  inverted, 

is  o:reater  than  2  ;  that  is,  that  -  4-->>2. 
^  h    ^   a 

11.  Show  that  a'^-{-h'^-\-c'^^ah-\-ac-\-hc,  unless  a=h=c. 

12.  If  x'^=a'^-^l'^,  and  i/'^z=c^-{-d'^,  which  is  greater,  xi/ 
or  ac-{-hd?  Ans.  xi/. 

13.  Show  that  ahc'^(a-]-h — c)(a-[-c — h)(h-\-c — a),  un- 
less a=:h=c. 

2d  Bk.  16 


186  RAYS  ALGEBRA,  SECOND  BOOK. 


YII.    QUADRATIC    EQUATIOIS^S. 

234.  A  Quadratic  Equation,  or  an  equation  of  the 
second  degree,  is  one  in  which  the  greatest  exponent  of  the 
unknown  quantity  is  2 ;  as,  x^-{-x=a. 

An  equation  containing  two  or  more  unknown  quantities, 
in  which  the  greatest  sum  of  the  exponents  of  the  un- 
known quantities  in  one  term  is  2,  is  also  a  Quadratic 
Equation  ;  as,  x2/=a,  xij — x — y=-c. 

225m  Quadratic  equations,  containing  only  one  unknown 
quantity,  are  divided  into  two  classes,  p«/-e  and  affected. 

A  Pure  Quadratic  Equation  is  one  that  contains  only 
the  second  power  of  the  unknown  quantity,  and  known 
terms;  as, 

x'^2=¥l—Ax\  and  ax''^h=cx'—d. 

A  pure  quadratic  equation  is  also  called  an  incomplete 
equation  of  the  second  degree. 

An  Affected  Quadratic  Equation  is  one  that  contains 
both  the  first  and  second  power  of  the  unknown  quantity, 
and  known  terms ;  as, 

hx'^-\-^tx=iZ^,  and  ax"^ — hx'^-\-cx — dx=e — f. 

An  affected  quadratic  equation  is  also  called  a  complete 
equation  of  the  second  degree. 

!336«  The  general  form  of  a  pure  equation  is  ax^=h. 

The  general  form  of  an  affected  equation  is  ax'^-\-hx=c. 

Every  quadratic  equation  containing  only  one  unknown 
quantity  may  be  reduced  to  one  of  these'  forms. 

For,  in  a  pure  equation,  all  the  terms  containing  x^  may 
be  collected  into  one  term  of  the  form,  ax^  j  and  all  the 
known  quantities  into  another,  as  h. 


QUADRATIC  EQUATIONS.  187 

So,  in  an  affected  equation,  all  the  terms  containing  x^ 
may  be  reduced  to  one  term,  as  ax^ ;  and  those  contain- 
ing X  to  one,  as  hx ;  and  the  known  terms  to  one,  as  c. 


PURE    QUADRATIC    EQUATIONS. 

22T. — 1.  Let  it  be  required  to  find  the  value  of  x  in 
the  equation,  }^x^ — 3-|-p\ic'^^12| — x^. 

Clearing  of  fractions,    4x2— 36-f5a:2=rl 53—1 2a:2; 
Transposing  and  reducing,  21a:2— 189; 

Dividing,  a:2=9; 

Extracting  the  square  root  of  both  members, 

X—zizZ]  that  is,  a:=r-|-3,  or  a:— —3. 

Verification.        1(^-3)2- 3+y'5^(-|-3)2=rl2|-(H  3)2. 

3— 3+3|=12|-9;  or33^3|. 

Since  the  square  of  —3  is  the  same  as  the  square  of  -f  3,    the 
value  x=z — 3,  will  give  the  same  result  as  x^=-f^. 

2.  Given  ax'^-\-h=id-\-cx?^  to  find  the  value  of  x. 

Transposing,       ....     ax'^—cx'^—d—b\ 
Factoring, {a  —  C)x'^  —  d — 6; 

r.'    '.'                                                                 2      ^-* 
Dividing, x^= ; 


>a— c 

From  the  preceding  examples,  we  derive  the  following 

Rule  for  the  Solution  of  a  Pure  Equation. — Reduce 
the  equation  to  the  form  ax^=:b.  Divide  hy  the  coefficient 
of  x^,  and  extract  the  square  root  of  both  members. 

228.  If  we  solve  the  equation  ax^=h,  we  have, 

X^=z±:\l-:  that  is,  X—-{-\-,  and  X— — \l~. 


188  RAYS  ALGEBRA,  SECOND  BOOK. 

The  equation  may  be  verified  by  substituting  either  of  these  values 
of  X.     Hence,  we  infer, 

1st.   That  in  every  "pure  equation  tlte  unJawwn  quantity  lias 
two  values^  or  roots,  and  only  two. 

2d.  Thxit  these  roots  are  equal  in  value,  hut  have  contrary 


1.  \\x^—U=h:c'^10 Ans.  a;=±3. 

2.  \(x^—Vl)=\x'—l Ans.  .-c^rtG. 

3.  (a;-f  2)^=4x+5 Ans.  x=^l 

4-  i--V  +  T-T9-==25 Ans.  ^-±.3. 

1 — Zx       l-\-lx 

_        X-\-l  X — 7  7  .  ,  o 

xi' — ^x      x'-\-lx      x' — 73 

6.   -^+-^z=c.  .     .     .  Ans.  x=^^x/h'c:'—2ahc. 

h-\-x       b — X  c  ^ 

1.  xi/6-\-x'=l-\-x'' Ans.  x^±^, 

2a.'  a      - 

8.  x-^./a'-irx'=---    ,—-—,.       .      .     .  Ans.  a:=±oi/3. 

2  2  

a — i/a'^ — x'^     ,  .  _,2ai/b 

10    ^  =^h.  .     ....       Ans.  a:=±-^-f^ . 

a-\-i/a'—x'  ^+1 


QUESTIONS    PRODUCING    PURE    EQUATIONS. 
!339.  For  the  statement  of  the  equation,  see  Art.  154. 

1,  What  two  numbers  have  the  ratio  of  2  to  5,  the  sum 
of  whose  squares  is  261  ? 

Let  2x  and  bx—  the  numbers. 

Then,  4x^-\-25x^=29x^=261 ; 

AVhence,  x^r^9,  and  x=S. 

Hence,  2X—Q,  and  5a:=:15    the  required  numbers. 


QUADRATIC   EQUATIONS.  189 

2.  The  square  of  a  certain  number  diminished  by  17,  is 
equal  to  130  diminished  by  twice  the  square  of  the  num- 
ber.    Required  the  number.  Ans.  7. 

3.  Required  a  certain  number,  which  being  subtracted 
from  10  and  the  remainder  multiplied  by  the  number  itself, 
gives  the  same  product  as  10  times  the  remainder  after 
subtracting  6|  from  the  number.  Ans.  8. 

4.  What  number  is  that,  the  J  part  of  whose  square 
being  subtracted  from  30,  leaves  a  remainder  equal  to  |  of 
its  square  increased  by  9  ?  Ans.  6. 

5.  There  are  two  numbers  whose  difference,  is  |  of  the 
greater,  and  the  difference  of  their  squares  is  128  ;  find 
them.  Ans.  18  and  14. 

6.  Divide  21  into  two  such  parts,  that  the  square  of  the 
less  shall  be  to  that  of  the  greater  as  4  to  25. 

Let  X  and  21— a:=  the  parts. 

Then,  rc2:  (21— a:)2  :  :4:25; 

Or,  (Auith.,  Art.  200,)  25a:2z=4(21— a:)^; 

Extracting  square  root,  5a:=2{21 — x)\ 

Whence,  aj=6,  and  21— a;=15, 

7.  Divide  14  into  two  such  parts,  that  the  quotient  of 
the  greater  divided  by  the  less,  shall  be  to  the  quotient  of 
the  less  by  the  greater,  as  16  to  9.  Ans.  6  and  8. 

8.  What  number  is  that  which  being  added  to  20  and 
subtracted  from  20,  the  product  of  the  sum  and  difference 
shall  be  319?  Ans.  9. 

9.  Find  two  numbers,  whose  product  is  126,  and  the 
quotient  of  the  greater  by  the  less  3.^  .      Ans.  6  and  21. 

10.  The  product  of  two  numbers  is  p,  and  their  quo- 
tient q.     Required  the  numbers.  .  —       ,      /» 

Ans.  |/p<2  and  ^r-. 

11.  The  sum  of  the  squares  of  two  numbers  is  370,  and 
the  difference  of  their  squares  20B-  Required  the  num' 
bers.  Ans.  9  and  17. 


100  RAY'S  ALGEBRA,  SECOND  BOOK. 

12.  The  sum  of  the  squares  of  two  numbers  is  c,  and 
the  difference  of  their  squares  d.     Required  the  numbers. 

Ans.  l\/'2{c^d),  and  iy2{c—d). 

13.  A  certain  sum  of  money  is  lent  at  5  %  per  annum. 
If  we  multiply  the  number  of  dollars  in  the  principal  by 
the  number  of  dollars  in  the  interest  for  3  mon.,  the  prod- 
uct is  720.     What  is  the  sum  lent?  Ans.  $240. 

14.  It  is  required  to  find  3  numbers,  such  that  the  prod- 
uct of  the  1st  and  2d  =a.,  the  product  of  the  1st  and  3d 
=6,  and  the  sum  of  the  squares  of  the  2d  and  3d  =^c. 

15.  The  spaces  through  which  a  body  falls  in  different 
periods  of  time,  being  to  each  other  as  the  squares  of  those 
times,  in  how  many  sec.  will  a  body  fall  through  400  ft., 
the  space  it  falls  through  in  one  sec.  being  16.1  ft.? 

Let  X=i  the  required  number  of  seconds. 

Then,  16.1  :  400  :  :  P  :  a;2  ;  whence,  a:=4.98-f-  sec. 

In  what  time  will  it  fall  1000  ft.?     Ans.  7.88+  sec. 

16.  What  two  numbers  are  as  3  to  5,  and  the  sum  of 
whose  cubes  is  1216? 

Let  3a:  and  bx—  the  numbers; 
Then,  27x3+125^:3==  152ic3=1216; 
Whence,  a:3=8,  and  a:=r^8=2. 
Hence,  the  numbers  are  6  and  10. 

This  is  properly  a  pure  equation  of  the  third  degree;  but  ques- 
tions producing  such  equations  are  generally  arranged  with  those 
of  the  second  degree. 

17.  A  money  safe  contains  a  certain  number  of  drawers. 
In  each  drawer  there  are  as  many  divisions  as  there  are 
drawers,  and  in  each  division  there  are  four  times  as  many 
dollars  as  there  are  di:awers.  The  whole  sum  in  the  safe 
is  $5324 ;  what  is  the  number  of  drawers?         Ans.  11. 


QUADRATIC  EQUATIONS.  191 

18.  A  and  B  set  out  to  meet  each  other ;  A  leaving  the 
town  C  at  the  same  time  that  B  left  D.  They  traveled  the 
direct  road  from  C  to  D,  and  on  meeting,  it  appeared  that 
A  had  traveled  18  miles  more  than  B;  and  that  A  could 
have  gone  B's  journey  in  15|  days,  but  B  would  have 
been  28  days  in  performing  A's  journey.  What  is  the  dis- 
tance between  C  and  D?  Ans.  126  miles. 

19.  Tw^o  men,  A  and  B,  engaged  to  work  for  a  certain 
number  of  days  at  different  rates.  At  the  end  of  the  time, 
A,  who  had  played  4  days,  received  ^5  shillings  ;  but  B, 
who  had  played  7  days,  received  only  48  shillings.  Had 
B  played  only  4  days,  and  A  7  days,  they  would  have 
received  the  same  sum.  For  how  many  days  were  they 
engaged?  Ans.  19. 

20.  A  vintner  draws  a  certain  quantity  of  wine  out  of  a 
full  vessel  that  holds  256  gal.  ;  and  then  filling  the  vessel 
with  water,  draws  off  the  same  number  of  gal.  as  before, 
and  so  on  for  fpur  draughts,  when  there  were  only  81  gal. 
of  pure  wine  left.    How  much  wine  did  he  draw  each  time? 

Ans.  64,  48,  36,  and  27  gal. 


AFFECTED    QUADRATIC    EQUATIONS. 

S30« — 1.  Required  to  find  the  value  of  x  in  the  equation, 

a.;_6a:+9=4. 

It  is  evident,  from  Art.  184,  that  the  first  member  of  this  equa- 
tion is  a  perfect  square.  By  extracting  the  square  root  of  both, 
members, 

We  find,     ....     a:— 3^dz2; 

Whence,     ....    a;=3 ±2=3 4-2=5,  or  3-2=1. 

Verificaiion.     (,5)2-6(5)-f  9=4;  that  is,  25— 30+9=4. 
(l)2_6(l)+9=4;  thfitis,    1—6+9=4. 

Hence,  x  has  two  values,  +5,  and  +1,  either  of  which  verifies  the 
equation. 


192  RAY'S  ALGEBRA,  SECOND  BOOK. 

2.  Required  to  find  the  value  of  x  in  the  equation, 

As  the  left  member  of  this  equation  is  not  a  perfect  square,  we 
can  not  find  the  value  of  X  by  extracting  the  square  root,  as  in  the 
preceding  example.  We  may,  however,  render  the  first  member  a 
perfect  square  by  adding  9  to  it. 

This  may  be  done  provided  the  same  number  be  added  to  the  other 
member,  to  preserve  the  equality.     The  equation  then  becomes, 
a:2_6a:4  9=36. 

Extracting  the  square  root,      a;— 3:=.=t:6. 

Whence,  :r=:3dt6^=-(  9,  or  — 3,  either  of  which  values  of  x  will 
verify  the  equation. 

!S31.  We  will  now  proceed  to  explain  the  method  of 
completing  the  square. 

Since  every  afi'ected  equation  (Art.  226)  may  be  reduced  to  the 
form, 

ax^-^hx^iC^ 

he 

Dividing  both  sides  by  a,       x^^—Xzzz—. 

b  c 

For  the   sake  of  simplicity,   let  — =2r),  and  ~—-Q.     As  each  of 

these  fractions  may  be  either  positive  or  negative,  the  equation  must 
assume  one  of  the  four  following  forms : 

x^-\-2px=q.  (1) 

x^—2px=q.  (2) 

x'^-\-2px=~q.  (3) 

x^—2px=—q.  (4)     Hence, 

Every  affected  equation  may  he  reduced  to  the  form 
x'^±2px=dzq. 

It  will  now  be  shown  that  the  first  member  of  this  equation  may 
always  be  made  a  perfect  square. 

We  may  consider  x'^-\-2px  as  the  first  two  terms  of  the  square  of 
a  binomial,  the  third  term  being  unknown  or  lost. 

Extracting  the  root  of  .r-,  we  find  that  the  first  term  of  the  bino- 
mial must  be  X.  We  next  observe  that  2px  is  twice  the  product  of 
the  first  term  by  (he  second;  therefore,  p,  which  is  half  the.  coeffi- 
cient of  Xj  is  the  second  term  of  the  binomial,  and  its  square,  p^^ 


.      QUADRATIC  EQUATIONS.  193 

added  to  x'^-\~2px,  will  render  it  a  perfect  square.     But,  to  preserve 
the  equality,  we  must  add  the  same  quantity  to  both  sides. 

This  gives,     .     .     .    x--^2px-{^p^=q-^p^j 

Extracting  the  square  root,    x-\'P=^zizy/q-'rp^', 
Transposing, X——pzh^q-^p'^. 

It  is  obvious  that  in  each  of  the  remaining  three  forms,  the  square 
may  be  completed  on  the  same  principle. 

Solving  equations  (2),  (3),  and  (4),  and  collecting  together  the 
four  diflferent  forms,  we  have  the  following  table: 


(1)  x^^2px=q.  x=—pdLi/q-\-p^. 

(2)  x^—2px=q.  a:=4-pzfc|/g+p2; 


(3)        x^-\-2px=—q.  x=—pdoi/—q-\-p^ 


(4)        x^-2px=—q.  x=-^pzti^  —q-\  p^. 

From  the  preceding  we  derive  the  following 

Rule  for  the  Solution  of  an  Affected  Equation.— 1st. 

Reduce  the  equation,  hy  clearing  of  fractions  and  transjwsi- 
iion,  to  the  form  ax^-|-bx=c. 

2d.  If  the  coefficient  of  x^  is  minus,  change  the  signs  of  all 
the  terms,  or  midtiply  each  term  hy  — 1. 

3d.  Divide  each  side  of  the  equation  hy  the  coefficient  of  x^. 

4th.  Add  to  each  memher  the  square  of  half  the  coefficient 
of  X. 

5th.  Extract  the  square  root  of  hoth  sides,  transpose  the 
known  term,  to  the  second  7nemher,  and  find  the  value  of  x. 

Remark. — Although  from  the  equation  rc^^m^,  we  have 
dba;=r=hm;  that  is,  -\-x=^m{\),  -\-x=—m{2),  —x=^m{^), 
and  — a;=— r?2(4),  it  is  evident  that  equations  (1)  and  (4)  are  the 
same  equation,  as  also  (2)  and  (3).  Hence,  -f  rr— ±m,  embraces  all 
the  values  of  x.  For  the  same  reason  it  is  necessary  to  take  only 
the  plus  sign  of  the  square  root  of  {x-\-pY. 

1.  Given  17a:— 2x^=32— 3.t,  to  find  x. 

Transposing, — 22:2-4- 20a:=  32; 

Changing  signs,  and  reducing,  a:^— 10a:— — 16; 
Completing  the  square  by  adding  (y))2=r25  to  both  sides, 
a:2_l0a:-(-25=— 16+25=9 ; 
2d  Bk.   :7- 


194  RAY  S  ALGEBRA,  SECOND  BOOK. 

Extracting  the  root,  a;— 5=  ±3; 
Whence,     ....  a:=5±3=8,  or  2. 

Verification.     17(8)— 2(8)2=32-3(8),  or  +  8=+  8. 
17(2)-2(2)2=32— 3(2),  or  +26=+26. 

2.  Given  3x^—2x^65,  to  find  x. 

Dividing  by  3,    .     .     .    .x'^~%x=^^; 

Completing  the  square,   x'^—%x^[^Y=^^^{lY=\^. 

Extracting  the  root,    .     a:~^:=rty. 

Whence, a:=^=h:L4^5,  or  —4^. 

Both  of  which  values  verify  the  equation. 

3.  Given  4.d'~2x' ^2ax=l^ah—lU\  to  find  x. 

Transposing,     .     .     .     — 2a;2-f  2aa:=— 4a2^18a6— 1862; 
Dividing  by -2,  .     .  x-—ax=^2a^—^ab^^b'^\ 

Completing  the  square,       x~—ax-\—T-=^—7 9a6-|-962; 

Extracting  root,    .     .  a:— ^=±1  -^ — 36  I; 

AVhence,  ^=?  +  (  T  ~^^  )=2a— 36,  or  —a-f  36. 

4.  Given  .'r-f-|/'(5cc-|-10)=8,  to  find  x. 

By  transposition,     .     .     |/(5a:-f-10)=8— a;; 
By  squaring,       .     .     .     5a:+10~64— 16a:+a;2; 

Or, a;2-21a:=— 54; 

Completing  the  square,    x'^-2\x-\-{'^^ )2=rl4 1  —54=226 ; 
Extracting  the  root,      .     a:— 2^1  =±1^5; 
Whence,  a:=2^i  _j_i^5^3^6^-18,  ^r  j=3. 

These  two  values  of  a:  are  the  roots  of  the  equation,  x"^ — 21a:=r — 54 
but  they  will  not  both  verify  the  original  equation. 

For,  the  proposed  equation  might  have  been  a:dr]/(5x-fl0)=8; 
and  the  operations  which  have  been  employed  would  result  in  the 
same  equation,  x'^ — 21a;=: — 54,  whether  the  sign  of  the  radical  part 
be  -|-  or  -'• 

Hence,  in  the  equation  X-\--y/{hx-\'\0)^=%,  the  value  of  a;  is  3 ;  but 
Id  the  equation  a:— j/(5a:4-10)=i8,  the  value  is  18. 


QUADRATIC  EQUATIONS. 


195 


5. 

6. 

7. 
8. 
9. 

10. 

11. 

12. 
13. 
14. 

15. 


x'—16x=—60. 

x'—6x=6x-^2S. 


10 


4-350— 12a:=^0 


2x=^+-..     .     . 

X 

3a:^+10a:=5V. 

(x— 1)(.7:— 2)=^!. 

ix^— ix+2^9. 
^  ,  110 


x-f22 
3~" 


4      Ox-6 


16. 

17. 

18.  l'7a:^4-19x=1848.  . 

19. 

20. 

21. 


3      ■10'^«~^" 
Sx^\x:'=10. 


22. 


^2_3,^  +  ^-.^4^     g^.- 

.T-f-4  1—X_4:X-\-7 

9 


23.  a:+- 


X — 3 

1 


^-^x     ^^x      13 
X 1 

X  X 


26.  •I  +  «-?=0. 
a      £c      a 


Ans.  x^=6,  or  — 10. 
Ans.  ic=10,  or  — 6. 
Ans.  x= — 6,  or  — 10. 
.  Ans.  x=io,  or  10. 
Ans.  £c=:14,  or  — 2. 

Ans.  x=:70,  or  50. 

.  Ans.  x=Sy  or  — 1. 

Ans.  x=S,  or  — 6J. 
Ans.  a;=^(3zt:^5). 
Ans.  a:=4,  or  — 3^. 

Ans.  a:=ll,  or  — 10. 

.    Ans.  x^=2,  or  ^|. 

Ans.  a:=3,  or  — 2|. 
Ans.  a:=9[f,  or  — 11. 


.  Ans.  z=l,  or  — i. 


Ans.  x=r-6dz2i/—l. 
Ans.  ic=:^4,  or  — 3|. 

Ans.  ic=21,  or  5. 

Ans.  a:=|/3,  or  l\/S. 

.     .  Ans.  x=S,  or  — |. 

Ans.  5c=l=±:|/(l— a'). 


196  RAY'S  ALGEBRA,  SECOND  BOOK. 

ZO.  ='zax — cx^ Ans.  a;= . 

c  c 

27.  x^ — (a-\-h~)x-\-ab=:^0.     .     .     .      Ans.  x=a,  or  h. 

28.  (a—h)x'—(a-\-h)x-^2b=0.    Ans.  x=l,  or  -— . 


29.  inqx"^ — ninx-\-pqx — jip=0.        Ans.  x=-,  or 
SO.  -^^-(a^-b^)x==  ^ 


m 


c^-\-b^  (ab')  ^-\-(a'b)   - 

Ans.  x=:a,  or  — b. 


31.  adx — acx'^=bcx — bd.      .     .     Ans.  x=~,   or . 

c  a 

12 

32.  |/ (0^4-5)=     /no-x-      •    A°s-  ^=4,  or  -21. 

«>«5. ;=  = 7=^ Ans.  a:r=4. 

4+1/^  1/^ 

34.  i/'s^—2i/X=X Ans.   X=z4:. 

35.  y^x-\-a — i/x-\-b=y^2x. 

Ans.  x=—'^zhli/2a'-\-2b\ 

Q^i       /-r-i     / ^-"^  A  4a         3a 

00.  ya-\-x-\-ya — x=  r~ — ■ — .     Ans.  ic:=-^,  or -=-. 

233.  Hindoo  Method  of  Solving  ftnadratics.— When 

an  equation  Is  brought  to  the  form  ax'^-\-bx=rCj  it  may 
be  reduced  to  a  simple  equation,  without  dividing  by  the 
coefficient  of  cc',  thus  avoiding  fractions. 

If  we  multiply  every  term  of  the  equation  ax^-l-bx=:C,  by  four 
times  the  coefficient  of  the  first  term,   and  add  to  both  sides  the 
square  of  the  coefficient  of  the  second  term,  we  shall  have 
4a2a:2-|-4a6a;+62^4ac-]-62 

Now,  the  first  member  of  this  equation  is  a  perfect  square,  and  by 
extracting  the  square  root  of  both  sides,  we  have 


2ax-\-b^zdc:y/4ac-\-b^,  which  is  a  simple  equation.     Hence,  the 


QUADRATIC  EQUATIONS.  197 

Hindoo  Eule  for  the  Solution  of  Quadratic  Equa- 
tions.— 1st.   Reduce  the  equation  to  the  form  ax^-fbx=rc. 

2d.   Multiply  both  sides  by  four  times  the  coefficient  of  x^. 

3d.  Add  the  square  of  the  coefficient  of  x  to  each  side, 
extract  the  square  root,  and  finish  the  solution. 

1.  Given  2a:2— 5a:=3,  ^o  find  x. 

Multiplying  both  sides  by  8,  four  times  the  coefficient  of  a:^, 

"Wehave 16a:2— 40a;=24. 

Adding  to  each  side  25,  which  is  the  square  of  the  coefficient  of  X, 

We  have     ....      16a:2—40a:+ 25=49; 

Extracting  the  root,       4a:— 5=  ±7;  whence,  a:=3,  or  — J. 

Find  the  value  of  the  unknown  quantity  in  each  of  the 
following  examples  by  the  Hindoo  Rule : 

2.  3x2+5a:=2 Ans.  x=\,  or  —2. 

3.  rr2+a:=30 Ans.  x=^b,  or  —6. 

4.  x^ — x=^2 Ans.  x=^9,  or  — 8. 

40        9*7 

5.  -^-f^=13 Ans.  x=^9,  or  i|. 

X 5  X  '13 

By  an  inspection  of  the  forms  given  in  Art,  231,  it  will  be  seen 
that  the  value  of  the  unknown  quantity  may  be  found  without  the 
formality  of  completing  the  square,  by  the  following 

Rule. — Reduce  the  equation  to  the  form  x^-f-^px^q.  The 
unknown  quantity  will  then  be  equal  to  one  half  the  coefficient 
of  its  first  power  taken  with  a  contrary  sign,  plus  or  minus 
the  square  root  of  the  square  of  the  number  last  written  to- 
gether with  the  known  quantity  in  the  second  member  of  the 
equation   taken  icith  its  proper  sign. 

Thus,  let  a:2-fl6a;=— 60.  _ 

Then,  a:=-8±:v/64-60=-8db/4=— 8dr2. 

a;=— 6,  or  —10. 
After  some  exercise  in  completing  the  square,  it  is  best  to  employ 
this  last  method. 


198  RAY'S  ALGEBRA,  SECOND  BOOK. 


PROBLEMS    PRODUCING     AFFECTED     EQUATIONS. 

2SSm — 1.  A  person  bought  a  certain  number  of  sheep 
for  $40,  and  if  he  had  bought  2  more  for  the  same  sum 
they  would  have  cost  $1  apiece  less.  Required  the  num- 
ber of  sheep,  and  the  price  of  each. 

Let  a;:=  the  number  of  sheep. 

40 
Then,  — =  the  price  of  one. 

40 

And         ^=  the  price  of  one,  on  the  second  supposition. 

m,        .  40        40      ,    ,       , 

Therefore,  — —^zr^  —  — 1,  by  the  question. 

Solving  this  eq.,  a;^— lzt:9=8,  or  — 10,  number  of  sheep. 

40  40       40 

And  — =iP=$5,  price  of  each.     Also,  —  =  — —  = — 4. 
X       ^  ^   X       — 10 

The  negative  value,  — 10,  to  fulfill  the  conditions  of  the  question 
in  an  arithmetical  sense,  must  be  modified,  on  the  principles  ex- 
plained in  Art.  164,  thus : 

A  person  tells  a  certain  number  of  sheep  for  $40.  If  he  had 
sold  2  fewer  for  the  same  sum  he  would  have  received  $1  apiece 
more  for  them.     Required  the  number  sold. 

2.  Find  a  number  such,  that  if  1*7  times  the  number 
be  diminish^!  by  its  square,  the  remainder  shall  be  *70. 

Let  a:=  the  number. 

Then,  \'ix—x'^=lQ. 

Or,  x'^~llx=—lQ. 

Whence,  a;^7,  or  10. 
In  this  case,  both  values  of  x  satisfy  the  question  in  its  arithmeti- 
cal sense. 

Thus,  17X  7—  72^.119-  49^70. 

Or,  17X10— 102:^170-100:.z:70. 

3.  Of  a  number  of  bees,  after  |,  and  the  square  root 
of  \  of  them,  had  flown  away,  there  were  two  remaining ; 
what  was  the  number  at  first? 


QUADRATIC  EQUATIONS.  199 

To  avoid  radicals,  let  2a:2  represent  the  number  of  bees  at  first; 

16x2 
Then,  — -  _[-a;+2=2a;2. 
y 

Whence,  X=:i6,  or  — 1^;  but  the  latter  value,  being  fractional, 
though  satisfying  the  equation,  is  excluded  by  the  nature  of  the 
question;  the  number  of  bees  is  2x62=72. 

4.  Divide  a  into  two  parts,  whose  product  shall  be  h'K 

Let  x=:  one  part ;  then,  a — a;=  the  other. 
Therefore,  x{a — x),  or  ax — x'^=Jfi. 
Whence,  a;=^(art^a2— 462) .  ^hat  is. 


x=^(a±i/a^—4b^),  and  a-Xz:^^{az:fzy/a^—Ab^),  are  the 
parts  required,  and  the  two  parts  are  the  same,  whether  the  upper 
or  lower  sign  of  the  radical  quantity  be  used.  Thus,  if  the  num- 
ber a  is  20,  and  b  8,  the  parts  are  16  and  4,  or  4  and  16. 

The  forms  of  these  results  enable  us  to  determine  the  limits  under 
which  the  problem  is  possible;  for  it  is  evident  that  if  462  i^q  greater 
than  a2^  j/ci2 — 462  becomes  imaginary/. 


The  extreme  possible  case  will  be,  when  i/a^ — 462:=^0,  in  which 
case  x—^a,  and  a—x=^a\  also,  62=zia2^  ^^d  b—^a. 

In  the  following  examples,  that  value  of  the  unknown  quantity 
only  is  given,  which  satisfies  the  conditions  of  the  question  in  an 
arithmetical  sense: 

5.  What  two  numbers  are  those  whose  sum  is  20  and 
product  36?  Ans.  2  and  18. 

6.  Divide  15  into  two  such  parts  that  their  product  shall 
be  to  the  sum  of  their  squares  as  2  to  5.  Ans.  5  and  10. 

7.  Find  a  number  such,  that  if  you  subtract  it  from  10, 
and  multiply  the  remainder  by  the  number  itself,  the  prod- 
uct shall  be  21.  Ans.  7  or  3. 

8.  Divide  24  into  two  such  parts  that  their  product  shall 
be  equal  to  35  times  their  difference.       Ans.  10  and  14. 

9.  Divide  the  number  346  into  two  such  parts  that  the 
sum  of  their  square  roots  shall  be  26.  Ans.  11^  and  15'^ 

SroGESTioN. — Let  a;=  the  square  root  of  one  of  the  parts,  and 
26— a;,  of  the  other. 


200  *        RAYS  ALGEBRA,  SECOND  BOOK. 

10.  What  number  added  to  its  square  root  gives  132? 

Aus.  121. 

11.  What  number  exceeds  its  square  root  by  48  J  ? 

Ans,  56J 

12.  What  two  numbers  are  those,  whose  sum  is  41,  and 
the  sum  of  whose  squares  is  901  ?  Ans.  15  and  26. 

13.  What  two  numbers  are  those,  whose  difference  is  8, 
and  the  sum  of  whose  squares  is  544?    Ans.  12  and  20. 

14.  A  merchant  sold  a  piece  of  cloth  for  $24,  and 
gained  as  much  per  cent,  as  the  cloth  cost  him.  Required 
the  first  cost.  Ans.  $20. 

15.  Two  persons,  A  and  B,  had  a  distance  of  39  miles 
to  travel,  and  they  started  at  the  same  time  ;  but  A,  by 
traveling  |  of  a  mile  an  hour  more  than  B,  arrived  one 
hour  before  him ;  find  their  rates  of  traveling. 

Ans.  A  3i,  B  3  mi.  per  hr. 

16.  A  and  B  distribute  $1200  each  among  a  number 
of  persons ;  A  gives  to  40  persons  more  than  B,  and  B 
gives  $5  apiece  to  each  person  more  than  A  ;  find  the 
number  of  persons.  Ans.  120  and  80. 

17.  From  two  towns,  distant  from  each  other  320  miles, 
two  persons,  A  and  B,  set  out  at  the  same  instant  to  meet 
each  other  ;  A  traveled  8  miles  a  day  more  than  B,  and 
the  number  of  days  before  they  met  was  equal  to  half  the 
number  of  miles  B  went  in  a  day ;  how  many  miles  did 
each  travel  per  day?  Ans.  A  24,  B  16  mi. 

18.  A  set  out  from  C  toward  D,  and  traveled  7  miles  a 
day.  After  he  had  gone  32  miles,  B  set  out  from  D  to- 
ward C,  and  went  every  day  -j'^  of  the  whole  journey;  and 
after  he  had  traveled  as  many  days  as  he  went  miles  in  one 
day,  he  met  A.     Required  the  distance  from  C  to  D. 

Ans.  VG,  or  152  miles. 

19.  A  grazier  bought  a  certain  number  of  oxen  for  $240, 
<ind  after  losing  3    sold  the  remainder  for  $8  u  head  more 


QUADRATIC  EQUATIONS.  201 

than    they  cost   him,    thus   gaining   $59    by  his   bargain. 
What  number  did  he  buy?  Ans.  16. 

20.  Divide  the  number  100  into  two  such  parts  that 
their  product  may  be  equal  to  the  difierence  of  their 
squares.  Ans.  38.197,  and   61.803  nearly. 

21.  Two  persons,  A  and  B,  jointly  invested  $500  in 
business  ;  A  let  his  money  remain  5  months,  and  B  only 
2,  and  each  received  back  $450,  capital  and  profit.  How 
much  did  each  advance?  Ans.  A  $200,  B  $300. 

22.  It  is  required  to  divide  each  of  the  numbers  11 
and  17  into  two  parts,  so  that  the  product  of  the  first 
parts  of  each  may  be  45,  and  of  the  second  parts  48. 

Ans.  5,  6,  and  9,  8. 

Represent  the  four  parts  by  x,  11 — re,  — ,  and  17 — — ,  and  put  the 
product  of  the  second  and  fourth  equal  to  48. 

23.  Divide  each  of  the  numbers  21  and  30  into  two 
parts,  so  that  the  first  part  of  21  may  be  three  times  as 
great  as  the  first  part  of  30  ;  and  that  the  sum  of  the 
squares  of  the  remaining  parts  may  be  585. 

Ans.  18,  3,  and  6,  24. 

24.  Divide  each  of  the  numbers  19  and  29  into  two 
parts,  so  that  the  difference  of  the  squares  of  the  first 
parts  of  each  may  be  72,  and  the  difference  of  the  squares 
of  the  remaining  parts  180.     Ans.  7,  12,  and  11,  18. 


DISCUSSION  OF  THE  GENERAL  EQUATION  IN  QUADRATICS. 

334.  The  discussion  of  the  general  equation  in  quad- 
ratics consists  in  investigating  its  general  properties,  and 
in  interpreting  the  results  derived  from  making  particular 
suppositions  on  the  different  quantities  which  it  contains. 

Taking  the  general  form,  (Art.  231,)  x--{'2px=q,  and  completing 

the  square,  we  have 

a;2-f2pa:-|-p2^g^p2. 


202  RAY  S  ALGEBRA,  SECOND  BOOK. 

Now,  x*-\-2pxJrP^={x-^pf.     For  the  sake  of  simplicity, 

Put,     ....     q-Jrp-=m^ ;  that  is,  yq^p^^m. 
Then,.     .     .     .     {x-^pf=m^; 
Transposing,     .     [x^p)2—m^=^0. 

But,  since  the  left  member  is  the  difference  of  two  squares,  it  may 
be  resolved  into  two  factors  (Art.  93);  this  gives 

{x-j^p-i-7n)(x-^p — m)=0. 

Now,  this  eqniation  can  be  satisfied  in  two  waj'S,  and  in  onli/  two; 
that  is,  by  making  either  of  the  factors  equal  to  0.  If  we  make  the 
second  factor  equal  to  zero,  we  have 

x^p—m—0; 
Or,  by  transposing,     x=—p^m=-p^y/q-]^p^. 

If  we  make  the  first  factor  equal  to  zero,  we  have 

x^  p^m^^O; 
Or,  by  transposing,  X:=. — p — nizzn—p—y/q-^p^.     Hence,  we  have 

Property  1st. — Every  quadratic  equation  has  fun  roofs, 
(or  values  of  the  unknown  quantity,)  and  only  two. 

From  the  equation  {x-\-p-{-m){x-{'P—Tn)—0,  we  derive 

Property  2d. — Every  affected  equation,  reduced  to  the 
form  x'^-{-2px=q,  may  he  decomposed  into  two  binomial 
factors,  of  which  the  first  term  in  each  is  x,  and  the  second, 
the  two  roots  with  the  siyns  changed. 

Thus,  the  two  roots  of  the  equation,  rc^— Ta^-j-lOi^O,  are  a;=2 
and  x=^b.     Hence,  x- — lx^\(i=[X — 2)(a:— 5). 

It  is  now  evident  that  the  direct  method  of  resolving  a 
quadratic  trinomial  into  its  factors,  is  to  place  it  equal  to 
zero,  and  then  find  the  roots  of  the  residting  equation. 

In  this  manner  let  the  trinomials,  Art.  94,  be  solved. 

By  reversing  the  operation,  we  can  readily  forni  nn  equa- 
tion whose  roots  shall  have  any  given  values.      Thus, 


QUADRATIC  EQUATIONS.  203 

Let  it  be  required  to  form  an  equation  whose  roots  shall 
be  — 3  and  4. 

We  must  have     .     .     .  x=—S,  or  a:+3=0, 

And x=4,  or  x—4:=zO. 

Hence, (a;+3)(a:— 4)=0; 

Or, a:2-a;-12=^0; 

Or, X^ — a;:^12,  which  is  an  equation  whose 

roots  are  —3  and  -)-4. 

1.  Find  an  equation  wliose  roots  are  4  and  5. 

Ans.  a;^— 9a:=— 20. 

2.  Whose  roots  are  — \  and  -j-s-         Ans.  cc^-f"6^=^6- 

3.  Find  an  equation,  without  fractional  coefficients,  whose 
roots  are  g  and  i.  Ans.  Ibx^ — 22a:= — 8. 

4.  Find  an  equation  whose  roots  are  m-{-n  and  m — n. 

Ans.  x'^ — 2mx=-u^ — 7n\ 

Resuming  the  equation  x'^-\-2px=q,  and  denoting  the 
two  roots  by  x'  and  x",  we  have 

x'  =-p+yq^^ 
x'^=—p—^q-^p'^. 

Adding,  x^-f  a:^'— _ 2p.  But,  — 2p  is  the  coefficient 

of  a:,  taken  with  a  contrary  sign.     Hence,  we  have 

Property  3d. —  The  sum  of  the  tico  roots  of  a  quadratic 
equation^  reduced  to  the  form  x^-)-2px=q,  is  equal  to  the 
coefficient  of  the  first  power  oj  x,  taken  with  a  contrary  sign. 

If  we  take  the  product  of  the  roots,  we  liave 

x^x^'=p-     ....     — [q^p'^)-=L — q. 
But  — q  is  the  known  term  of  the  equation,  taken  with  a  con- 
trary sign.     Hence,  wc  have 

Property  4th. —  Tlie  product  of  the  two  roots  of  a  quad- 
ratic equation^  reduced  to  the  form  x'^-(-2px=q,  is  equal  to 
the  known  term  taken  ivifh  a  contrary  sign. 


204  RAY'S  ALGEBRA,  SECOND  BOOK. 

In  the  preceding  demonstrations,  we  liave  regarded  2p  and  q  as 
both  positive;  but  the  same  conclusions  will  be  obtained  by  taking 
them  both  negative,  or  one  positive  and  the  other  negative. 

2Sf%m  We  shall  now  consider  the  essential  sign,  and 
numerical  magnitude,  of  the  roots  in  each  of  the  four 
forms. 

It  is  evident  that  the  value  of  \/q-rP^  must  be  greater  than  JD, 
since  the  square  root  of  p"^  alone,  is  p. 

But  the  value  of  y' — q^p^  must  be  less  than  p,  since  it  is  the 
square  root  of  a  quantity  less  than  p^. 

With  these  principles,  a  careful  consideration  of  the  roots,  or 
values  of  x  in  each  of  the  four  different  forms,  will  render  the  fol- 
lowing conclusions  evident: 

1st  form, rc2-j-2pa:=g. 

The  first  root  is  essentially  positive,  the  second  essentially  nega- 
tive ;  and  the  first  is  numerically  less  than  the  second. 

2d  form, x^ — 2px=zq. 

x^=p-{^yq-HP^,  and  x^^=p—i/q^p^. 

The  drst  root  is  essentially  positive,  the  second  essentially  nega- 
tive; and  the  first  is  numerically  greater  than  the  second. 

3d  form, x^-{-2px——q. 

x'=.  'P-\-\/—q-\-p'-,  and  x^^——p—  yZ-q -\-p^. 

Both  roots  are  essentially  negative,  and  the  first  is  numerically 
less  than  the  second. 

4th  form, x^—2px= — q. 

X^—p-\- 1/ — g-f p2,  and  x'^:=p — |/ — q-k-p"^. 

Both  roots  are  essentially  positive,  and  the  first  is  numerically 
greater  than  the  second. 

!330.  We  shall  now  proceed  to  phow  wlien  the  roots 
become  imaginary,  and  why. 

In  the  third  and  fourth  forms,  the  radical  part  is  y/ — <?-fp^. 
Now,  when  q  is  greater  than  p-,  this  is  essentially  negative,  and 
the  extraction  of  the  root  is  impossible,  (Art.  193.)     Hence, 


QUADRATIC  EQUATIONS.  205 

When  the  known  term,  is  negative,  and  greater  thari  the 
square  of  half  the  coefficient  of  the  first  power  of  x,  the  roots 
are  imaginary. 

To  show  why  the  roots  are  imaginary,  we  must  prove  that 

When  a  number  is  divided  into  two  equal  parts,  their 
product  is  greater  than  that  of  ajiy  other  two  parts  into 
which  the  number  can  be  divided. 

Or,  as  the  same  principle  may  be  otherwise  expressed, 

The  product  of  any  tico  uneqzial  numbers  is  less  than  the 
square  of  half  their  sum. 

Let  2p  represent  any  number,  and  let  the  parts  into  which  it  is 
supposed  to  be  divided,  be  p-{-z  and  p — z.  The  product  of  these 
parts  is 

Now,  this  product  is  evidently  the  greatest,  when  2^  jg  the  least; 
that  is,  when  Z^^z^O,  or  Z=^0.  But  when  z  is  0,  the  parts  are  p  and 
p,  which  proves  the  proposition. 

Now,  it  has  been  shown,  (Art.  234,  Properties  3d  and  4th,)  that 
2p  is  equal  to  the  sum  of  the  two  roots,  nnd  that  q  is  equal  to  their 
product.  But,  when  q  is  greater  than  p^,  we  liave  the  pioduct  of 
two  numbers  greater  than  the  square  of  half  their  sum,  which,  by 
the  preceding  principle,  is  impossible. 

If,  then,  any  problem  furnishes  an  equation  of  the  form 
X~ziz2px:^ — q,  in  which  q  is  greater  than  p^,  the  conditions  are 
incompatible  with  each  other.     The  following  is  an  example: 

Let  it  be  required  to  divide  the  number  8  into  two  parts, 
whose  product  shall  be  18. 

Let  z  and  8—x  represent  the  parts. 
Then,  x{8-x)=18;  or  a:2_8a:=— 18; 
Whence,  a:=4-!-/~2,  or  4—/— 2. 

These  expressions  for  the  values  of  X,  show  that  the  problem  is 
impossible,  which  is  obviously  true.  By  the  preceding  theorem,  the 
greatest  product  of  the  parts  of  8  is  10. 


206  RAY'S  ALGEBRA,  SECOND  COOK. 

13S7*  Examination  of  particular  cases. 

1st.  If,  in  the  third  and  fourth  forms,  where  q  is  negative,  we 
suppose  q—'P^y  the  radical,  ^Z— Q'-fjp^  becomes  0,  and  a:=— p  in 
one,  and  +P  i^  the  other. 

It  is  then  said,  iht  two  roots  are  equal. 

In  fact,  if  we  substitute  p'^  for  g,  the  equation  in  the  3d  form 
oecomes 

Hence,  {^-^-pf,  or,  {x-^p){x-\-p)—0. 

The  first  member  is  the  product  of  two  equal  factors,  either  of  which, 
placed  equal  to  zero,  gives  the  same  value  for  x.  A  like  result  is 
obtained  by  substituting  p^  for  q  jn  the  fourth  form. 

2d.  If,  in  the  general  equation,  x'^-\^2px=^q,  we  suppose  Q'— 0, 
the  two  values  of  x  reduce  to, 

x=~p^p^O,  and  x=—p—p=—2p. 
In  fact,  the  equation  is  then  of  the  form 

x^-\-2px^0,  or  x{x^2p))=.Q, 
which  can  be  satisfied  only  by  making 

X—{),  ora;-f2p=:^0;  whence,  a;r=0,  or  Xz= — 2p. 

3d.  If,  in  the  general  equation,  x'^-^2px^^q,  we  suppose  2p^0, 
we  have 

x^—q-,  whence,  a:=:=hyg. 

In  this  case,  the  two  values  of  x  are  equal  and  have  con- 
trary signs,  real,  if  q  is  j^ositive,  as  in  the  first  and  second 
forms,  and  imaginary,  if  q  is  negative,  as  in  the  third  and 
fourth  forms. 

Under  this  supposition  the  equation  contains  only  two  terms,  and 
belongs  to  the  class  treated  of  in  Art.  228. 

4th.  If  2p=^0,  and  (7=0,  the  equations  reduce  to  iC^^rO,  and  give 
the  two  values  of  X,  in  all  the  forms,  each, equal  to  0. 

S3S*  There  remains  a  singular  case  to  be  examined, 
which  is  sometimes  met  with  in  the  solution  of  problems 
producing  quadratic  equations. 

To  discuss  it,  take  the  equation  ax'^-\-hx=c. 


QUADRATIC  EQUATIONS.  207 

Solving  this  equation,  the  values  of  x  are 


^-  2a  '    ^-  2d 

If,  now,  we  suppose  a^^O,  these  values  become 
_6  1  6     0         —6—6      -26 

That  is,  one  value  of  x  is  indeterminate  and  the  other  infinite. 
(Arts.  136,  137.) 

But  if  we  suppose  a=0  in  the  given  equation,  we  have 

bx—C,  and  x——. 
b 

We  now  propose  to  show  that  the  indeterminate  value  is  the  same 
as  the  one  last  found,  and  that  the  infinite  value  simply  expresses  an 
impossibility. 

If  we  multiply  both  terms  of  the  second  member  of  the  equation 

—  6-f-v'6^+4ac   ,         ,        -^ — i — 
x=^ — \ — ,  by  —6 — ^b^-\-4ac,  we  have 

62_(624  4ac)  — 4ac 


2a(— 6— ^  6^-t4acj      2a(— 6— ,/ 6^+ 4ac)' 

Or,  by  dividing  both  terms  by  2a,  and  making  a::=0, 

—2c               —2c      c 
/jj— -__ _^^  — _ 

_6— ,/ 62-f-4ac      —26      6' 

0  .  c 

Hence,  we  see,  that  the  value  of  ^  —  7.,  is  really  ^,  and  arises  from 

having  made  the  fiictor,  2a,  equal  to  zero.     (See  Art.  136.) 

By  supposing  tt—O,  the  equation  ax^->rbx=:C,  reduces  to  bx—C, 

an  equation  of  the  first  degree,  which  can  have  but  one  root. 

The  supposition  that  it  has  two,  gives  one  value  infinite,  which  is 

equivalent  to  saying,  the  equation  has  but  one  finite  root. 

If  we  had  at  the  same  time  a=iO,  6^0,  C— 0,  the  equation  would 

be  altogether  indeterminate.     This  is  the  only  case  of  indetermina- 

tion  occurring  in  quadratic  equations, 

'     S39.  We  shall  now  apply  the  principles  above  stated, 
in  the  discussion  of*  the  followin"; 


208  RAY'S  ALGEBRA,  SECOND  BOOK. 

Problem  of  the  Lights. — It  is  required  to  find,  in  a 
line  BC,  which  joins  two  lights,  B  and  C,  of  different  in- 
tensities, a  point  which  is  illuminated  equally  by  each. 


P^^  B  P         C  P^ 

It  is  a  principle  in  optics,  that  the  intensity  of  the  same 
light  at  different  distances,  is  inversely  as  the  square  of  the 
distance. 

Let  a  be  the  distance  BC  between  the  two  lights. 

Let  b  be  the  intensity  of  the  light  B  at  the  distance  of  1  ft.  from  B. 

Let  C  be  the  intensity  of  the  light  C  at  the  distance  of  1  ft.  from  C. 

Let  P  be  the  point  required. 

Let  BP  =X]  then,  CP  =a—x. 

By  the  principle  above  stated,  since  the  intensity  of  the  light  B 
at  the  distance  of  1  foot,  is  6,  at  2,  3,  4,  .     .     .     .    feet,  it  must  be 

A-  7{-  ir.     •     •     •     •      )    hence,  the  intensity  of  B  and  of  C,  at  the 
4   9   16                                                              be 
distance  of  x  and  of  a — x  feet,  must  be    -tt  and .,. 

'  x^  [a — x)^ 

But,  by  the  conditions  of  the  problem,  these  two  intensities  are 
equal.     Hence,  we  have  for  the  equation  of  the  problem, 

which  reduces  to 


x'^     {a—xf  x^         6' 

Whence,    =:: — — ,  or  — ^_. 

^  ^b  |/6 

This  gives  the  following  results  : 

,  ay/lb         ,  awTi 

1st.  x=i--zi^ — -]  whence,  a — x=—:^- — :::., 

2d.  x=  —^ — -;  whence,  a—x=      .,  '^  -. 
|/6— |/c  y/b—yc 

"We  shall  now  proceed  to  discuss  these  values. 


QUADRATIC  EQUATIONS.  209 

I.  Let  h^c. 

CLr/'  b 
The  first  value  of  X,      — =1,    is    positive,  and   less    than  a,  for 

^       -  is  a  proper  fraction.    Hence,  this  value  gives  for  the  point 


illuminated  equally,  a  point  P  situated  between  B  and  C.  We  per- 
ceive, also,  that  the  point  P  is  nearer  to  C  than  B ;  for,  sincv 
&>c,    we    have    i/6+/6>y/6+|/c^    or    '2i/byy^-\-y^e,    and 

—z£z =:>2)  ^'^d)  consequently,    ■ — ^ =">  — 

This  is  manifestly  correct,  for  the  required  point  must  be  nearer 
the   light    of  less   intensity.      The    corresponding    value   of    a — a:, 

(Xi/C        .  Ob 

— :=^ — -   IS  positive,  and  evidently  less  than  tt^. 
/6+^c       ^  '  ^  2 

The  second  value  of  x,    -  .- — -,  is  positive,  and  greater  than  a ; 

yb—yc 

This  value  gives  a  point  P^,  situated  on  the  prolongation  of  BC, 
and  in  the  same  direction  from  B  as  before.  In  fact,  since  the  two 
lights  emit  rays  in  all  directions,  there  will  be  a  point  P'',  to  the 
right  of  C,  and  nearer  the  light  of  less  intensity,  which  is  illumin- 
ated equally  by  the  two  lights. 

The  second  value  of  a—x,  -^~— ^,  is  negative,  as  it  ought  to 
yb—x/c 
be,  and  represents  the  distance  CP'',  in  the  opposite  direction  from  G, 

(Art.  47.) 

II.  Let  5<c. 


P^^  B         P  C  P^ 

The  first  value  of  x^       ^  Y -,  is  positive,  and  less  than  — ,  for 

_      _      _     —^  i/6"  aWb        a 

vi>+vo>vbH^b;  •••,75q:;7s<i,and-^-5^3<2. 

2,1  Bk.  18 


210  KAYS  ALGEBRA,  SECOND  BOOK. 


The  corresponding  value  of  a — x,  — -zJ- — -,  is  greater  than  ~^' 

and  positive.  These  values  of  a:,  and  a— rr,  show  that  the  point  P 
is  situated  between  B  and  C,  but  nearer  to  B  than  C.  This  is  evi- 
dently a  true  result,  since,  under  the  present  supposition,  the  intens- 
ity of  the  light  B  is  less  than  that  of  the  light  C. 

The  sscond  value  of  a:,   — ^=f^ — -    q^  — _        _,  is  essentially  neg« 

ative,  and  represents  a  point  P^'',  in  the  opposite  direction  from  B. 
As  the  intensity  of  the  light  B  is  now  supposed  to  be  less  than  that 
at  C,  there  is,  obviously,  another  such  point  of  equal  illumination. 

The  corresponding  value  of  a — a;,  is  — r:^ — -^=^ — —- — =.    It  is 


|/6— |/c      yc 


1  c~ 


positive  and  greater  than  a,  for   t/c>i/c— i/6   .-.  — = x^l, 

y  0 — j/  o' 

and  — =^ !>«.     This  represents  CP^^,  and  is  the  sum  of  the  dis- 

|/c— 1/6 

tances  CB  and  BP^'',  in  the  same  direction  from  C  as  before. 

m.  Let  h=c. 

The  first  values  of  x  and  of  a — x,  reduce  to  ^,  which  shows  that 

the  point  illuminated  equally  is  at  the  middle  of  the  line  BC,  a  re- 
sult manifestly  true,  upon  the  supposition  that  the  intensities  of  the 
two  lights  are  equal. 

The   other   two    values    are    reduced    to      ^    =00  .     (Art.  136.) 

This  result  is  manifestly  true,  for  the  intensities  of  the  two  lights 
being  supposed  equal,  there  is  no  point  at  ^ny  finite  distance,  except 
the  point  P,  which  is  equally  illuminated  by  both. 


IV.  Let  l=c,  and  a=0. 

The  first  system  of  values  of  x  and  a — a:,  become  0.  This  is  evi- 
dently correct,  for  when  the  distance  BC  becomes  0,  the  distances 
BP  and  CP  also  become  0. 

The  second  system  of  values  of  x  and  a — X.  become  ^;  this  is  the 

symbol  of  indetermination,  (Art.  137.) 


QUADRATIC  EQUATIONS.  211 

This  result  is  also  correct,  for  if  the  two  lights  are  equal,  and 
placed  at  the  same  point,  every  point  on  either  side  of  them  will  be 
illuminated  equally  by  each. 

V.  Let  ci=0,  h  not  being  =c. 

All  the  values  of  a;  and  a — X  reduce  to  0;  hence,  there  is  no  point 
equally  illuminated  by  each.  In  other  words,  the  solution  of  the 
problem  fails  in  this  case,  as  it  evidently  should. 

This  might  also  have  been  inferred  from  the  original  equation ; 

for  if  we  put  a=Q,  -^5=7 r,  becomes  -^  =  -^,  which  can  never 

^  'a:2      (re— a)2  x^     x^' 

be  true  except  when  6=c,  as  in  Case  IV. 

SSO"*.  Examples  for  discussion  and  illustration. 

1.  Required  a  number  such,  that  twice  its  square,  in- 
creased by  8  times  the  number  itself,  shall  be  90. 

Ans.  5,  or  — 9. 

How  may  the  question  be  changed,  that  the  negative  answer, 
taken  positively,  shall  be  correct  in  an  arithmetical  sense? 

2.  The  diiference  of  two  numbers  is  4,  and  their  prod- 
uct 21.     Required  the  numbers. 

Ans.  -f-3,  +7,  or  —3  and  —7. 

3.  A  man  bought  a  watch,  which  he  afterward  sold  for 
$16.  His  loss  per  cent,  on  the  first  cost  of  the  watch,  was 
the  same  as  the  number  of  $'s  which  he  paid  for  it.  What 
did  he  pay  for  the  watch?  Ans.  $20,  or  $80. 

4.  Required  a  number  such,  that  the  square  of  the  num- 
ber increased  by  6  times  the  number,  and  this  sum,  in- 
creased by  7,  the  result  shall  be  2.  Ans.  x= — 1,  or  — 5. 

What  do  the  values  of  a;  show?  How  may  the  question  be  changed 
to  be  possible  in  an  arithmetical  sense? 

5.  Divide  the  number  10  into  two  such  parts,  that  the 
product  shall  be  24.  Ans.  4  and  6,  or  6  and  4. 

Is  there  more  than  one  solution?     Why  ? 


212  RAY'S  ALGEBRA,  SECOND  BOOK. 

6.  Divide  the  number  10  into  two  such  parts  that  the 
product  shall  be  26.        Ans.  S-j-y — 1,  and  5 — -y/ — 1. 

What  do  these  results  show? 

7.  The  mass  of  the  earth  is  80  times  that  of  the  moon, 
and  their  mean  distance  asunder  240000  miles.  The  at- 
traction of  gravitation  being  directly  as  the  quantity  of 
matter,  and  inversely  as  the  square  of  the  distance  from 
the  center  of  attraction,  it  is  required  to  find  at  what  point 
on  the  line  passing  through  the  centers  of  these  bodies, 
the  forces  of  attraction  are  equal. 

Ans.  21o86o.5-[-  miles  from  the  earth, 
and  24134.5—  ''  "  "  moon. 
Or,  270210.44-  "  "  "  earth, 
and     30210.4 -f      "      beyond  the  moon  from  the  earth. 

This  question  involves  the  same  principles  as  the  Problem  of  the 
Lights,  and  may  be  discussed  in  a  similar  manner.  The  required 
results,  however,  may  be  obtained  directly  from  the  values  of  Xj 
page  208,  calling  a=240000,  6=80,  and  c=l. 


TRINOMIAL    EQUATIONS. 

S40.  A  Trinomial  Equation  is  one  consisting  of  three 
terms,  the  general  form  of  which  is  ax"^-\-bx'*=:c. 

Every  trinomial  equation  of  the  form 

that  is,  every  equation  of  three  terms  containing  only  two 
powers  of  the  unknown  quantity,  and  in  which  one  of  the 
exponents  is  double  the  other,  can  be  solved  in  the  same 
manner  as  an  affected  equation. 

As  an  example,  let  it  be  required  to  find  the  value  of  x 
in  the  equation 

X* — 2})x'^=q. 


QUADRATIC  EQUATIONS.  213 

Completing  the  square,     a:<— 2pa:2-^p2_g_|_p2^ 


x-— p=v'g-fp2. 


a;2=+p±/g+p2. 


241.  Binomial  Surds.  —  Expressions  of  the  form 
A±j/B,  like  the  value  of  x^  just  found,  or  of  the  form 
|/A±|/B,  are  called  Binomial  Surds. 

The  first  of  these  forms,  viz.,  A=t]/B,  frequently  re- 
sults from  the  solution  of  trinomial  equations  of  the  fourth 
degree ;  and  as  it  is  sometimes  possible  to  reduce  it  to  a 
more  simple  form  by  extracting  the  square  root,  it  is  neces- 
sary to  consider  the  subject  here. 

We  shall  first  show  that  it  is  sometimes  possible  to  ex- 
tract the  square  root  of  A±|/B,  or  to  find  the  value  of 


V 


A±|/B. 


Let  us  inquire  how  such  binomial  surds  may  arise  from 
involution. 

If  we  square  2=t|/3,  we  have  4±4|/3-j-3,  which,  by  reduction, 
becomes  7dt4|/3.     Hence,  ^7d=4|/3=2=ii:p/3.     In  the  same  way 

it  may  be  shown  that  -«(5dt2|/6=|/2db|/3. 

It  thus  appears  that  the  form  Adzy/B  may  sometimes  result  from 
squaring  a  binomial  of  the  form  azfc|/6,  or  |/a=irj/6,  and  uniting 
the  extreme  terms,  which  are  necessarily  rational,  into  one.  In 
such  cases,  A  is  the  sum  of  the  squares  of  the  two  terms  of  the  root, 
and  y^B  is  twice  their  product. 

To  find  the  root,  therefore,  put  x^-^y^—A  and  2a:?/^|/B,  and  pro- 
ceed to  find  X  and  y,  the  terms  of  the  root.     Thus, 

Extract  the  square  root  of    .     .     .     7-f4j,  3. 

Put x^^y^=7  (1),  and 

2xy^AyyS: 
Adding,  we  have         a;2-f  2:ri/4-2/2=7+4/37 
Subtracting,  we  have  x^— 2x7/-|-?/2— 7— 4/37 


214  RAY'S  ALGEBRA,  SECOND  BOOK. 


Extracting  the  root,       iC-j-^^^7-|-4|/3         (2). 


^-2/=v 


7_4,/3         (3). 


Multiplying  (2)  by  (3),  x^-y^=^^d—48=yl=l         (4). 

By  adding  and  subtracting  (1)  and  (4),  we  have  2x^=^8  .-,  x=2 
and  2y^=z6  .-.  y^zy'W.     Hence,  2-|-^3  is  the  root  to  be  found. 


1.  Extract  the  square  root  of  15-f-6|/6.     Ans.  3-f-|/6. 

2.  Of  34~24i/2 Ans.  4-3|/2: 

3.  Of  14±4i/6. Ans.  y2ziz2y'W. 

We  shall  now  proceed  to  demonstrate  more  fully  that 
the  square  root  of  A=t|/B  may  always  be  found  in  a  simple 
form,  when  A^ — B  is  a  perfect  square.  To  do  this  it  is 
necessary  to  prove  the  following  theorems : 

Theorem  I. — The  value  of  a  quadratic  surd  can  not  he 
partly  rational  and  partly  irrational. 

For,  if  possible,  let  |/'a:=a-f|/6;  ,•.  squaring  both  sides, 

_    X a"^ b 

x=a^-\'2ayb-{-b]  .-.  yb= ^ -;  that  is,  an  irrational 

quantity  is  equal  to  a  rational  quantity,  which  is  impossible. 

Theorem  II. — In  any  equation  of  the  form  xd:r|/y=:azt: 
-j/'^b,  the  rational  quantifies  on  opposite  sides  are  equal,  and 
also  the  irrational  quantities. 

For  if  X  does  not  =a,  let  x=ra-|-m; 

Therefore,  a-f  m+y/^— a+^/fi;  .-.  m-{-j/y=y/W; 
that  is,  the  value  of  a  quadratic  surd  is  partly  rational  and  partly 
irrational,  which  has  been  shown  by  Th.  T,  to  be  impossible ;  hence, 
x=a,  and  y/y=y''b. 

We  shall  now  proceed  to  find  a  formula  for  extracting 
the  square  root  of  A-f  |/B. 


QUADRATIC  EQUATIONS.  215 

Assume     ....    'JA.-{-■^/'B=-^/x^■^/'^, 

A-]-'i/h'=^x-^y-\-2-i/xy,  by  squaring. 
By  Th.  II,  a:+^=A(l);  and  2^^=/B(2); 
Squaring  equations  (1)  and  (2),  we  have 

Axy        — B; 


Subtracting,  a:2_2a;y+2/^=A2— B;  or,  (a;— i/)2=:A2— B. 


Let  A2— B  be  a  perfect  square  =C2;  then,  C=i/ A2 — B. 

Therefore,     .     .     (a;— ^)2— C^,  or  x—y~^\ 
But,     ....  iC-fy=A; 

Whence,  .     .     .    Xz= — ^ —  ;  and  2/= — ^r— . 


__  /A  +  C  _  /A— C 

And    .    .      i/a:=±'Y     2     I  ^'^^i  |/2/==t  \~2~'' 


Therefore,  i/aj+v^y,  or  ^A+i/B  =  rt^/A+2±^ 


a-4:j 


Similarly,  V^^W.  or  ^/A-^B=d=^4L+^:^  ^^^. 
Or,       .     .^^W^=^[^^^^\     (K.) 


And  .Va-./B=±(^/^_V^).     (L.) 

By  substituting  particular  values  for  the  general  ones  in  these 
formulas,  examples  may  be  easily  solved. 

1.  Extract  the  square  root  of  31-|-10|,/6. 

Here,    A=r31,    /B==10/6;    .-.  A2— B=C2z=961— 600rr:361; 
and  C=19. 

.-.  A+C:=50,    A— Cr=12. 


216  RAYS  ALGEBRA,  SECOND  BOOK. 

Taking  the  formula  and  substituting,  we  have 


Va^(V^%/-^). 

Proof  .— (5+ i/6)2^25+10|/ 6+6=31 +10/6. 


2.  Reduce  ^  np-\-2m^ — 2my'np-\-m\  to  its  simplest  form. 

Here,  A=np-}-2m^,  and  B^=i4'm^{np-\-m^). 

A2_B=in2p2^  and  C=np,  (formula  L.) 
.-.  A+C=2rip+2m2,  A— C=2m2  .-.  x—npJf-m^,  y=m^. 
Formula  (L)  gives  =h(|/^p+w2— m). 

3.  Find  the  square  root  of  ll  +  6|/2r     Ans.  3-f  |/2 

4.  Of  8±2|/2 Ans.  y  2±1 

5.  Of  l7+2i/60 Ans.  2|/3+|/5 

6.  Of  cc— 2;,/^^ Ans.  i/^Hl—l 

1.  Of  2a^/^l.     (A.^.0.)  .     Ans.  |/a(l-f  ^=1) 

8.  Of  x-\-j/-\-z-\-2i/xz-\-yz.      .     Ans.  yx-{-y-{-y^z, 


9.  Find  the  value  of  ^28+10|/3+^67-16i/3. 

Ans.  13. 

When  A2 — B  is  not  a  perfect  square,  or  when  the  binomial  surd 
is  of  the  form  |/Adt|/B,  the  root  will  be  more  complex  than  the 
original  form. 

Remark. — By  the  above  method  the  square  root  of  any  bino- 
mial or  residual,  as  a  +  6,  a— 6,  a'^-\^b^,  etc.,  may  also  be  found,  in 
a  complex  form. 

S4!3.  We  shall  now  resume  the  subject  of  Trinomial 
Equations.  The  general  form  of  trinomial  equations  is 
x^-\-2px'*=.q ;  but  there  are  several  varieties  of  this  form, 


QUADRATIC  EQUATIONS.  217 

of  which  the  following  are  the  principal :  viz.,  x-\~-i/ x=^q, 

p.  an 

x*-\-px^=q,    x'^-\-px'^:=q,    x^^-{-2)X^=q,    x''"-}-p^""=9'5     (j^^ 
-\-px-\-qy-\-h(x'^-\-px-\-q')=r,  and  (x'^-\-2JX-\-qy~"-\-h(x'^-\~2^^ 

+qy=k. 

Some  of  these  varieties,  if  developed,  would  produce  very  compli- 
cated expressions,  yet  they  may  all  be  solved  by  the  general  method 
given  in  Art.  240. 

1.  Given  x^ — 6x^^16,  to  find  the  value  of  x. 
Assume,     ....     X^—y;  then,  X^=7/-,  and 

Whence,     .     .     .     .     2/  ~8,  or  — 2. 

Therefore,      .     .     .     x3=8,  or  —2;  and  x^2,  or  —fZ 

Or,  the  example  may  readily  be  solved  without  introducing  a  new 
letter.     Thus, 

Completing  the  square,     x'' — 6a:'^-f  9=25. 
Extracting  the  root,  x^ — 3=  ±5 

a;3=:8,  or  —2,  and  x=2,  or  —f2. 

It  will  be  shown  hereafter,  (Art.  396,)  that  in  such  examples  as 
the  preceding,  there  are  four  values  of  X  not  deteimined.  ^ 

2.  Given  5.c — 4|/cc=33,  to  find  the  value  of  x. 

Assume,     ....     ^'x=^y;  then,  x=z^y^,  and 

52/2-42/=33; 
Whence,     ....     y— 3,  or — 'i; 
Consequently,    .     .     X—9,  or   ^X^. 


3.  Given  i/x-\-12-\-fx-\-12=G,  to  find  the  value  of  x. 


Assume,     f/x-\-12=y;  then,  -i/x-^12=y^,  and 

2/2+2/^6;   whence,  y=2,  or— 3; 
Therefore,  ^a;-fl2=2,  or  —3. 
Whence,     X-{-12^'[^  or  81 ;  and  X=.i,  or  69. 
2d  Bk.  19* 


218  RAY'S  ALGEBRA,  SECOND  BOOK. 

Or,  without  introducing  a  new  letter  2/,  we  may  proceed  to  com- 
plete the  square.     Thus, 


l/a;+12+f/a:-i-12+i=6+i=:2_5; 


-2' 


^a;+12=-^±|=+2,  or  -3. 
a:+12=:16,  or  81. 
Whence,  a:=4,  or  69. 


4.  Given  Sx'-\-i/Sx''-^l=bb,  to  find  the  value  of  x. 
Adding  1  to  each  member,  the  equation  becomes 


3a:2-^l+^3a:2+l=56. 
The  equation  may  now  be  solved  like  the  preceding. 
The  values  of  x  are  +4,  —4,  +|/2l,  and  —^21. 

Find  the  values  of  x  in  each  of  the  following  examples  : 

5.  x'—2bx'=—14:4:.     .     .     .  Ans.  x==:±^S,  or  ±4. 

6.  bx*-^1x'^67S2.  Ans.  x=±:6,  or  ±-pL|/=3740. 

7.  9x«— ll;r^.==488.  .     .      Ans.  x^2,  or  ^1^=183. 

8.  x3_aji^i5500.  .     .     Ans  x=:2b,  or  (—124)^. 

9.  x^4-x^'=i056.    .     .     .  Ans.  x=64,  or  (—33)^ 


10.  x-{-b=i/x-{-b-\-6.   .     .     .       Ans.  x=4:,  or  — 1. 

11.  2|/x'-^— Sx-f-ll^x^— 3.X+8.  

Ans.  x=.2,  1,  or  ^±A/^31. 

12.  x'^— 7x4-^/^^— Yx+18=-24.  

Ans.  x=9,  —2,  or  Klz^ylTS). 

13.  (x^— 9)2==3+ll(x2— 2).      Ans.  a;=±5,  or  ±2. 

Ans.  a;=4,  2,  or  ^(— 7±|/lV.) 
15.  .T*(l-h^|-(3:.H^)-^0. 

Ans.  x^S,  — 3^,  or  J(— 1±^/_251.) 


QUADRATIC  EQUATIONS.  219 

16.  x/(  l-^)  =  ^-.    Ans.  x=±^a±iy2). 

Sometimes  it  may  be  necessary  to  substitute  a  new  letter  two  or 
more  times,  or  to  complete  the  square,  without  substitution,  three  or 
more  times.     The  following  is  an  example: 


17.  x*-i-bx'-\-4i/x*-\-bx'=:60 


Ans.  x=.±2,  ±3^/-!,  and  ±-^|(— 1±:T/177) 

S43.  Equations  sometimes  occur  in  which  the  com- 
pound term  is  not  at  first  apparent,  but  which  may  be 
reduced  to  the  form  of  a  trinomial  equation  by  the  follow- 
ing method : 

Extract  the  square  root  to  two  or  three  terms,  and  if  we  find  a 
remainder  (omitting  known  terms,  if  necessary,)  which  is  any  mul- 
iiple  or  any  part  of  the  root  already  found,  the  given  equation  may 
"be  reduced  to  a  trinomial,  of  which  the  co7npound  term  will  be  the 
root  already  found. 

If  the  greatest  exponent  of  the  unknown  quantity  be  not  even, 
it  must  be  made  so  by  multiplying  both  members  of  the  equation  by 
the  unknown  quantity. 

-to      4 

1.  Given  x^ — 4:ax^ — 2a'^x-^12a^= ,  to  find  x. 

X 
Multiplying  botlj  sides  by  X,  and  transposing,  we  have 

Proceelling  to  extract  the  square  root,  we  have  the  following 

OPERATION. 

x*—Aax^    2a-x--\-12a^x—16a^x-—2ax. 


X* 


2x^    2ax\  — 4aa;3    2a^x^ 
-  4ax?'{-4a'^X" 
Remainder,  .     .     —6a^x^ -^12a^x     IGa^; 
Or, —6a^{x^—2ax)     16a*. 

Ilencc,  the  given  equation  may  be  written  thus: 

[x^    2axY—Ga^{x^—2ax)—16a*^0. 
Or,  .    .    .     {x-—2ax)^—6a'2{x^-2ax)^16a*. 
Proceeding  with  the  solution,  we  find 

X— 4a,  —2a,  or  adra^/"^. 


220  RAY'S  ALGEBRA,  SECOND  BOOK. 

2.  x*— 2a;3— 2a:2+3xr=108. 

Ans.  a;:=4,  —3,  or  -'(Irfc^/ZISS). 

3.  x'—2a^-\-x=S0.     Ans.  x=S,  —2,  or  i  (Idzy-HTO). 

4.  x'— 6a:^+lla:— 6=0.     .     .     .     Ads.  x=1,  2,  or  3. 

5.  x*—Qx^-{^bx'^12x=60.  

Ans.  x=b,  —2,  or  i(3±^^=T5). 

6.  x*—Sx'-j-10x'-\-24:x=—b. 

Ans.  a;:=;5,    — 1,   or    2±|/5. 

7.  4x*4-'^=4x3+33.  Ans.x=2,  — ^,  or  j(l±y^=43). 

14     ta;^"^      3x     ~2x'^   ^' 

Ans.  x=i,  3,  or  -^(7^^/69). 


SIMULTANEOUS  QUADRATIC  EQUATIONS  CONTAINING  TAVO 
OR  MORE  UNKNOWN  QUANTITIES. 

S44.  ftuadratic  Equations,  containing  two  or  more 
unknown  quantities,  may  be  divided  into  two  classes,  pwre 
and  affected. 

Pure  Equations  embrace  those  that  may  be  solved  with- 
out completing  the  square. 

Affected  Equations  embrace  those  in  the  solution  of 
which  it  is  necessary  to  complete  the  square. 

The  same  equations  may  sometimes  be  solved  by  both 
methods. 

PURE    EQUATIONS. 

24:5m  Pure  equations  may  in  general  be  reduced  to  the 
solution  of  one  of  the  following  forms,  or  pairs  of  equa- 
tions. 

ao^tp}.  (2-)^Tp}.  c^-);!::::}. 


QUADRATIC  EQUATIONS.  221 

We  shall  explain  the  general  method  of  solution  in  each 
of  these  cases. 

To     solve     x^y=^a     (1),     and     xy^=h     (2),    we    must 
find  X — y. 

Squaring  Eq.  (1),  .  .  x'^^'tjcy-Yy'^=a^\ 
Multiplying  Eq.  (2)  by  4,  \xy  =r46; 
Subtracting,  .  .  .  x-—lxy^y'^—a?—^b, 
Or, (x-?/)2=:a2_45. 


AVhence,      ....  x-y=riz^a^—Ab; 

But, x^y=a\ 


Adding,  and  dividing  by  2,  X=^\a:±L\^ a^ — 46. 

Subtracting,  and  dividing  by  2,  y^=z\az^\^/ aP-^^h. 

The  pair  of  equations  (2)  is  solved  in  the  same  manner,  except 
that  in  finding  iC-|-?/,  we  must  add  4  times  the  second  equation  to  the 
square  of  the  first. 

The  pair  of  equations  (3)  is  solved  merely  by  adding  and  sub- 
tracting, then  dividing  by  2  and  extracting  the  square  root. 

1.  Given  x^^y'^=l^,  and  rc-f^=7,  to  find  x  and  y. 
Squaring  the  2d  Eq.,     a;2-f  2a:^-f-2/^=:49; 

But, a;2  +?/2=rr25        (1). 

Subtracting,     ...  2a:?/         =^24,        (2). 

Taking  (2)  from  (1),     x^~~1xy^-y'^=  \ 
Whence,       ....     x—y^r±z\  (3). 

But, x^y=l  (4). 

Adding  and  subtracting  (3)  and  (4),  and  dividing  by  2, 
a:=4,  or  3;  and  y—^,  or  4. 

2.  Given  x^-\-xy-\-y''=^\{^'),  and  a:+|/^-f 7/==13(2), 
to  find  X  and  y. 

Divide  Eq.  (1)  by  (2),      a:-|/^-f-2/=  7.  (3). 

But, a:+i/^-f2/=13.  (2). 

By  subtracting,  .     .     .         2|/'S^^6. 

Whence, ^xy—3,  and  xy—9.      (4). 


222  RAYS  ALGEBRA,  SECOND  BOOK. 

By  adding  [2)  and  (3),      .     .     .     x-^y^  10.       (5). 
Squaring,  (5),       .     .     .      a:2-f  2a2/+^2^100; 
Multiplying  (4)  by  4,    .  4x1/         ^  36; 

x2—2xi/^-y^-^  64,  .-.  x-y=±:8. 
But,  a:+7/=10;  whence,  X—9,  or  1 ;  and  2/=l,  or  9. 

Equations  of  higher  degrees  than  the  second,  that  can  be  solved 

by  simple  methods,  are  usually  classed  with  pure  equations  of  the 
second  degree. 

1  1  3  3 

3.  Given  x'^-\-y^=^6,  and  x^-\-y^=^126,  to  find  x  and  y. 

In  all  cases  of  fractional  exponents,  the  operations  may  be  simpli- 
fied by  making  such  substitutions  as  will  render  the  exponents  in- 
tegral. To  do  this,  put  the  lowest  power  of  each  unknown  quantity  equal 
to  the  first  power  of  a  new  letter. 

11  3-3 

In  this  example,  let  x^—V,  and^^=Q;  then,  a;4— p3  and  y^=Q\ 
The  given  equations  then  become, 

P-fQ=    6  (1), 

ps-f  Q3^126  (2). 

Dividing  Eq.  (2)  by  (1),  P2—  PQ-f  Q2z=21; 
Squaring  Eq.  (1),  .  .  P2+2PQ+Q2.^36; 
Subtracting,  ....  3PQ=15,  .-.  VQ=5. 

Having  P-|-Q=:6,  and  PQ=5,  by  the  method  explained  in  form 
(1),  we  readily  find  P^=5,  or  1;  and  Q=:l,  or  5. 

Whence,  a:=625,  or  1 ;  and  y=^l,  or  3125. 

4.  Given  (a^y)(x'—f)=160        (1), 

lx-{-7/Xx''-\-f)=bS0  (2),  to  find  X  and  y. 

ajS — x^—xy--]-y^=160     (1),  by  multiplying. 
a^-\-x^y-\-xy^-\-y^=580     (2),    "  « 

2x^y ^2x1/^=420     (3),  by  subtracting. 
Add  (3)  to  (2),     rc-H 3a:22/4-3tri/2+y3^1000. 
Extract  cube  root,      .     .     .     x-\-y—lO. 
From  (3),     ....      a:?/(a:4?/)-_210;  .-.  a:?/=21. 

From  x-\-y—10,  and  X1/—21,  we  readily  find  a:—?,  or  3;  and 
2/=3,  or  7. 


QUADRATIC  EQUATIONS. 


223 


Solve  the  following  by  the  preceding  or  similar  methods  : 


5.  x—y=2, 

6.  a:^+/=13,| 

xy  =   ^.) 

7.  ^xJry=1. 

4x'-^y'=2b. 

8.  x^-f=.16, 1  . 


x-y=2. 


9.  x+y=  11,) 
x'-\-y'=40l.  3 

/  10.   1(a^-\-y')=9(x'—y') 
x^y — y'^x=lQ. 

11.  x'-{-xy=Si,  )        . 
x'—y'  =24.  I       . 

12.  x'-^f=lb2,       ) 
sy     x'-xyJ^f=19.\ 


13.  x'-\-y'-\-xy=20S, 
x  -^y  =16. 

14.  x^—if=7xy,)       . 
^_3^_2.      I       . 

15.  a:*+a;y+y=91, 

16.  x—y=^x-^y^y, 

3  3 

a:-— 7/^=37. 

11.xi-^yi=   5, 
J-fy^=13. 

18.    05^3/'=    5, 

a:  -[-y  =35. 


Ans.  33=15,  or  — 13; 
3/=13,  or  — 15. 

Ans.  a;=±3; 

y=±2. 

Ans.  a;=:2,  or  | ; 
t/=S^  or  4. 

.     .      Ans.  x=5; 

Ans.  x=7,  or  4; 
^=4,  or  7. 

.     .      Ans.  £c=4; 
.    .  y=2. 

Ans.  a:;==t7 ; 
y=±5. 

Ans.  a;:=5,  or  3; 
y=S,  or  5. 

Ans.  a;=:12,  or     4; 

^=  4,  or  12. 

Ans.  a;=4,  or  — 2; 
3/=2,  or  —4. 

Ans.  icz=:rh3,  or  ±1 ; 
3/=dtl,  or  ±8. 

Ans.  a:=16,  or     9; 

y=  9,  or  16. 

Ans.  a;=16,  or  81; 
y=2l,  or     8. 

Ans.  a:=   8,  or  27; 
y=27,  or     8. 


224 


RAYS  ALGEBRA,  SECOND  BOOK. 


19.  x^-\-y-^=.   4, 

20.  x'^2f=Zhl, 

xy=^    14. 


21.  x-^y=   4,)    . 

22.  x{y^z)=a,  ^ 


Ans.  a;r=9,  or  1  ; 

?/=l,  or  9. 

Ads.  x=^, 


9. 


3/=2,  or  7. 


Ans.  x=^S,  or  1 ; 
y=l,  or  3. 


Ans  ^— f-J(^+^-^)(^+^-^) 
Ans.a:_=i=^         2(^>-j-c-a)         ' 

/(a-f?>-c)(6+c-«) 
^-     \  2(a-fc— ^.)         ' 

^_±  /(64-c-«)(«+^-Z') 


2(a-f;>— c) 


AFFECTED    EQUATIONS. 

S46.   The  most  general   form  of  quadratic  equations, 
^  "  containing  two  unknown  quantities,  is 

ax^  -\-hxy-\-cx-\-  dy"^  -\-  ey  -\-f=-  0 . 

By  arranging  the  terms  according  to  the  powers  of  x^ 
and  dividing  by  the  coefficient  of  the  first  term,  two  quad- 
ratic equations  containing  two  unknown  quantities,  may  be 
reduced  to  the  following  forms : 

a:^+(ay+&  )x-^^cf^dy-^^e  r^O  (1), 

x^J^{a:yJ^y)x^c'f^d'y^e'=.^         (2). 

To  find  the  values  of  either  of  the  unknown  quantities,  we  must 
eliminate  the  other.  We  shall  now  show  that  this  operation  pro- 
duces an  equation  of  the  fourth  degree. 

By  subtracting  the  second  equation  from  the  first,  and  making 
a—a'—a^^,  h—b'=b'\  etc.,  we  have 


Whence,  X— 


QUADRATIC  EQUATIONS.  225 

As  this  value  of  x  contains  ?/2,  that  of  x^  will  evidently  con- 
tain y*,  which  value  of  a:^,  substituted  in  the  first  equation,  neces- 
sarily gives  rise  to  an  equation  of  the  fourth  degree.     Hence, 

The  solution  of  two  quadratic  equations,  containing  two 
unknoicn  quantities^  depends  upon  the  solution  of  an  equation 
of  the  fourth  degree^  containing  one  unknown  quantity. 

As  there  are  no  direct  methods  of  solving  equations  of 
any  higher  degree  than  the  second,  those  of  the  class  now 
under  consideration  can  not  be  solved  except  in  particu- 
lar cases,  and  then  only  by  indirect  methods,  or  special 
artifices. 

We  now  proceed  to  point  out  some  of  these  special  cases, 
in  addition  to  those  already  referred  to  in  Arts.  242,  243, 
and  245,  with  some  of  the  more  common  artifices  em- 
ployed. 

S47.  There  are  two  cases  in  quadratics  which  may 
always  be  solved  as  equations  of  the  second  degree,  viz. : 

Case  I. — When  one  of  the  equations  rises  only  to  the 
first  degree. 

Given  ax-\-hy=c     (1), 

dx^-\-exy-\-fy'^-\-gx^hy=k     (2),  to  find  x  and  y. 

From  eq.  (1),  we  may  obtain  a  value  of  X  in  terms  of  y.  Sub- 
stituting this  value,  for  x  and  X-  in  (2),  the  new  equation  will  evi- 
dently contain  only  y  ^^^  V'- 

Case  II. — When  both  equations  are  homogeneous.  (See 
Art.  30.) 

Given     ax^-['h  xy-]^cy'^=:^d      (1), 

a'x^-\-h'xy-[-c'y^=^d'     (2),  to  find  x  and  y. 

Put  y^=^tx,  where  ^  is  a  third  unknown  quantity,  termed  an 
auxiliary  quantity.  Substituting  this  value  of  y  in  the  two  equa- 
tions, we  have 

a  0:24. h  tx'^^c  t^x'^=x'^{a  +6  t^c  t'^)^d         (3), 
a'xi-Yb'tx2-^&t'^x^~-^--x'^{a'^b't^&f^)^d'        (4). 


226  RAY'S  ALGEBRA,  SECOND  BOOK. 

From  eq.  (3),  we  find       .     .     .   x'^= j- -,  (5). 

From  eq.  (4),  we  find       ...   x^=-^^,jj-^^      (6). 

^,       .                                   d                    d' 
Therefore,    ....     j-: -r,—    ,     ,,^ 77,. 

Or,      ....    d{a'+b't^&t'^)=d\a^bt-Ycr-), 

a  quadratic  equation,  from  which  the  value  of  t  may  he  found, 
(Art.  281.)  and  thence  x  from  (5)  or  (6),  and  y  from  the  equa- 
tion yr=dx. 

248.  When  two  quadratic  equations  are  symmetrical 
with  respect  to  the  t^wo  unknown  quantities ;  that  is,  when 
the  two  unknown  quantities  are  similarly  involved,  they 
may  frequently  be  solved  by  substituting  for  the  unknown 
quantities  the  sum  and  diflference  of  two  others. 

1.  Given      ,  x  -\-y  =za     (1), 

x^-\-y^^=h     (2),  to  find  x  and  y. 

Let  x=iS-\-z,  and  y-S—z\  then,  s=^         (3), 

x^=s^^^s^z^\Os-'z'^-^\Os^z^-YhHZ^-\-z\ 
7/'  ^  .S'^'  _5.s<2;+10.s'32;2_  lOs'^z^^^az^-  z^; 

x'^-{-y->^2s^-\-20s^z^-\-10sz*=b. 
By  substituting  the  value  of  s—-^,  and  reducing,  we  find 

a2  2_166— o-'i 

^  "^2^  ~     80a    • 

Completing  the  square,  wc  find  the  value  of  Z]  and  from  (3),  that 
of  X  and  y. 

349.  An  artifice  that  is  often  used  with  advantage, 
consists  in  adding  such  a  number  to  both  members  of  an 
equation  as  will  render  it  a  trinomial  equation  that  can 
be  resolved  by  completing  the  square,  (Art.  240).  The 
following  is  an  example: 


QUADRATIC  EQUATIONS.  227 

x'-\-f=20  (2),  to  find  X  and  y. 

Since  /  -+^  \  =^-^2+^;  add  2  (o  each  side  of  eq.  (1),  and 
then  ^  to  complete  the  square. 

Whence,   - +'^=±3— 1:=.5    or  —  g. 

Let    ^+M;    then,    ^i+-<or?*=J. 

Whence,  a:?/=8,  and  2xi/=16. 

From   the   equation  a:2-)-?/2_20j    and    2a-?/=16,    we   readily  find 

ic=±4,  and  i/==h2. 

iC      ?/ 
By  taking   -  -f-    = — ?     two    other    values   of  x  and  2/  may  be 

found. 

S50.  It  is  often  of  advantage  to  consider  the  sum, 
difi"erence,  product,  or  quotient  of  the  two  unknown  quan- 
tities as  a  single  vnknowti  quantify,  and  find  its  value. 
Thus,  in  example  9,  following,  the  value  of  xy  should  be 
found   from    the  first   equation,  and   in    example    10,  the 

value  of  -. 

y 

Other  auxiliaries  and  expedients  may  frequently  be  em- 
ployed with  advantage,  but  their  use  can  only  be  learned 
by  experience,  judgment,  and  tact. 

Note. — In  some  of  the  examples  all  the  values  of  the  unknown 
quantities  are  not  given;   those  omitted  are  generally  imaginary. 

3.  x^^fJrX-^y=Z^O,  I    .     . 


X' — y'^-{-x — ^=1 50. 

X  +4y==14,  j 

/+4..:=23/+ll.  j 


Ans.  rc=15,  or 

-16; 

y=   9,  or 

-10. 

.  Ans.  x:=2,  or 

-46; 

?/=3,  or 

15. 

228 


RAY'S  ALGEBRA,  SECOND  BOOK. 


5.  2i/—Sx=U,              1 

Sx^-^2(y-iiy=u.i 

6.  x-~ij=2,      ^       .     .     . 

.     .     .     Ans.  x=   2,  or  ll; 
.     .     .               ?/=10,  or  84. 
.      .      .  Ans.  .T=5,  or          |; 

--^-1-'-   ( 

.     .     .            ^=3,  or  -11. 

7.  Sx'~^  xi/=lS,\      .    . 

4y-h3a:y=54.  j        .     . 

.    Ans.  x=d^2,  or  ±2^/3; 
y^±3,  or  qp3i/3. 

V     8.  x^+xy:=10,  1     .      .     . 
0:^+2/^24.  J     .     .     . 

.    Ans.  x=±2,  or  ±5|/2^ 
y=d=3,  or  =f4|/2. 

9.  4xi/=96-xy,l      .    . 
x^y=.6.           j       .     . 

^                   :.~y^2.          3    .     .     . 
")''  ll.  a.y=.180-8.r^,|     .      . 
^^^    ^  ^   ^+3y=ll.         j    .    . 

Ans.  a;r=2,  4,  or  3±^2l  ; 
y=4,  2,  or  3q=|/21. 

.     .     .  Ans.  a;=5,  or       j? ; 

•     •              y=3,  or  —  ,3^. 
.     .     .     .    Ans.  .T=5,  or  6; 
.     .     .     .             ^=2,  or  |. 

^^     12.  x-^y-i-^x+y=12,l    . 

x^-{-f=n.          J   . 

.     .     .     .  Ans.  .T=5,  or  4; 
.     .     .     .             i/=4,  or  5. 

13..a:+3^+^^+/=:.18,|       . 

Ans.  rr==3,  2,  or  — 3^1/3"; 
^=2,  3,  or  ~3zFy  3. 

14.  x'-\-SxJ^i/=lS-2xi/,') 
f-^Sy-{.x=.U.           1 

.     .       Ans.  .'c=4,  or  10; 

.     .               ?/=:_5,  or  —7. 

15.  a:^+a:/^12,)      .     .     . 
a:+a:/=:18.]      .     .     . 

.     .      Ans.  x=2,  or  16; 
.     .                y=2,  or     ^. 

16.  x'^f—x—i/=1S,')      . 
x-\-i/+xi/=^B9.       j      . 

.     .     .    Ans.  x=^d,  or  3; 
.     .     .             7/:-3,  or  9. 

)         Ans.  x=   6,  or  — 4.1 ; 
\                  y=^\2,  or  —9. 

18.       ^^       I  l/-^+-y-       ^' 
x=.f+2. 

L,")           Ans.  a:=r6,  or  3; 
J                      y=2,  or  1. 

QUADRATIC  EQUATIONS.  229 


QUESTIONS    PRODUCING    SIMULTANEOUS    QUADRATIC 

EQUATIONS  INVOLVING  TWO  OR  MORE 

UNKNOWN  QUANTITIES. 

25X» — 1.  There  are  two  numbers,  whose  sum  multi- 
plied by  the  less  is  equal  to  4  times  the  greater,  but 
whose  sum  multiplied  by  the  greater  is  equal  to  9  times 
the  less.     What  are  the  numbers  ?      Ans.  3.6,  and  2.4. 

2.  There  is  a  number  consisting  of  two  digits,  which 
being  multiplied  by  the  digit  in  the  ten's  place,  the  prod- 
uct is  46  ;  but  if  the  sum  of  the  digits  be  multiplied  by 
the  same  digit,  the  product  is  only  10.  Required  the 
number.  Ans.  23. 

8.  What  two  numbers  are  those  whose  difference  multi- 
plied by  the  difference  of  their  squares  is  32,  and  whose 
sum  multiplied  by  the  sum  of  their  squares  is  272? 

Ans.  5  and  3. 

4.  The  product  of  two  numbers  is  10,  and  the  sum  of 
their  cubes  133.    Required  the  numbers.    Ans.  2  and  5. 

Note. — The  preceding  problems  may  be  solved  by  pure  equations. 

5.  What  two  numbers  are  those  whose  sum  multiplied 
by  the  greater  is  120,  and  whose  difference  multiplied  by 
the  less  is  16?  Ans.  2  and  10. 

6.  Find  two  numbers  whose  sum  added  to  the  sum  of 
their  squares  is  42,  and  whose  product  is  15. 

\/  Ans.  3  and  5. 

7  Find  two  numbers  such,  that  their  product  added  to 
their  sum  shall  be  47,  and  their  sum  taken  from  the  sum 
of  their  squares  shall  leave  62.  Ans.  5  and  7. 

8.  Find  two  numbers  such,  that  their  sum,  their  product, 
and  the  difference  of  their  squares,  shall  be  all  equal  to  each 
other.  Ans.  :^-\.^y^b,  and  |-f  ^^5. 


230  RAY'S  ALGEBRA,  SECOND  BOOK. 

9.  Find  two  numbers  whose  product  is  equal  to  the  dif- 
ference of  their  squares,  and  the  sum  of  whose  squares  is 
equal  to  the  difference  of  their  cubes. 

Ans.  -j|/5,  and  |(5-[-|/ 5). 

10.  A  and  B  gained  by  trading  $100.  Half  of  A's 
stock  was  less  than  B's  by  $100,  and  A's  gain  was  -^^  of 
B's  stock.  Supposing  the  gains  in  proportion  to  the  stock, 
required  the  stock  and  gain  of  each. 

Ans.  A's  stock  $600,  B's  $400; 
A's  gain  $60,  B's  $40. 

11.  The  product  of  two  numbers  added  to  their  sum 
is  23 ;  and  5  times  their  sum  taken  from  the  sum  of  their 
squares  leaves  8.     Required  the  numbers.    Ans.  2  and  7. 

12.  There  are  three  numbers,  the  difference  of  whose 
differences  is  5  ;  their  sum  is  44,  and  continued  product 
1950;  find  the  numbers.  Ans.  25,  13,  6. 

252.  FormulaB.— A  General  Solution  to  a  problem 
producing  a  quadratic  equation,  like  one  in  simple  equa- 
tions, gives  rise  to  h  formula,  (Art.  162,)  which  expressed 
in  ordinary  language,  furnishes  a  rule.  We  shall  illustrate 
the  subject  by  a  few  examples. 

Express  each  of  the  following  formulae  in  the  form  of  a 
rule,  and  solve  the  numerical  example  by  it : 

1.  Investigate  a  formula  for  finding  two  numbers,  x  and 
y,  of  which  the  sum  of  their  squares  is  s,  and  difference 
of  the  squares  d. 

Ans.   x-^iy2(^:^);    ^J=li/W^)' 

Example. — Find  two  numbers  such  that  the  sum  and 
difference  of  their  squares  are  respectively  208  and  80. 

Ans.  12  and  8. 

2.  Investigate  a  formula  for  finding  two  numbers,  x  and 
y,  of  which  the  difference  is  c?,  and  the  product  p. 


QUADRATIC  EQUATIONS.  231 

Ex. — A  man  is  8  years  older  than  his  wife,  and  the 
product  of  tlie  numbers  expressing  the  age  of  each  is 
2100.     How  old  are  they?  Ans.  Man  50,  wife  42. 

3.  Investigate  a  formula  for  finding  a  number,  cc,  of 
which  the  sum  of  the  number  and  its  square  root  is  s. 

Ans.  x=^s-\-l—i/s-\-l. 

Ex. — The  sum  of  a  number  and  its  square  root  is  272 ; 
what  is  the  number?  Ans.  256. 

4.  The  same  when  the  difference  of  the  number  .r,  and 
its  square  root  is  cZ.  Ans.  x=d-\-\-^y  d-{-\. 

Ex. — Find  a  number  such  that  if  its  square  root  be  sub- 
tracted from  it,  the  remainder  will  be  132.     Ans.  144. 

5.  Given  x-\-i/=s^  and  xy^=p,  to  find  the  value  of 
ir?-\-ij\  x*-\-y^j  and  x'^-^-y*^  in  terms  of  s  and  p. 

Ans.  x^-\-y^z=s^ — 2p; 
x?-{-y^=ii^ — ^ps ; 
x^J^y^^&'—^ps'^^pK 

Ex. — The  sum  of  two  numbers  is  5,  and  their  prod- 
uct 6.  Required  the  sum  of  their  squares,  of  their  cubes, 
and  of  their  fourth  powers.  Ans.  13,  35,  and  97. 

253.  Special  Solutions  and  Examples. — If  an  equa- 
tion can  be  placed  under  the  form 

(ic+a)X=.0, 

in  which  X  represents  an  expression  involving  a-,  at  least 
one  value  of  the  unknown  quantity  may  be  found. 

For  since  the  equation  will  be  satisfied  by  making  either 
factor  =0,  we  have  x-\-a=i()^  and  X^O.  Therefore, 
a;= — a,  is  one  solution  of  the  equation,  and  the  other 
^^alues  of  X  will  be  found  by  solving,  if  possible,  the  equa- 
tion X=:0. 

Thus,  the  equation  x"^ — x-—^x-\A:=0,  may  be  placed  under  the 
form  (re— 2)(a:2-fa:— 2)=0.  Hence,  x—2^0,  or  x=-\-2;  and,  from 
the  other  factor,  we  find  a:::=-f  1,  or  — 2. 


232  RAY'S  ALGEBRA,  SECOND  BOOK. 

Skill  in  separating  such  an  equation  into  its  factors  must  be 
acquired  by  practice. 

2 

1.  Given  x — 1=2-| ~,  to  find  x. 

yx 

2         2 
Since  x—l^{Wx-^\){y'X—].)  and  2+—^=    ^ (i/^+l); 

y  X     yx 

_  _  2       _ 

Therefore,  {^/x^l){yx-l)=^—=~{yx-^\); 

y  X 

Therefore,   ^/^+l^-0,  and  a;=(— 1)2=1. 
2 

Also,    y/rc— 1  =  — =,  by  dividing  by  y^+1. 

Whence,    \^X—2,  or  —1;  and  rc--r4,  or  1. 

2.  a;^ — Sx=2.      (x\dd  2.x  to  each  side.) 

Ans.  x= — 1,  or  2. 

^'  ^^-£^^^-         (  Transpose   i  and  |.  )         ^ 
Ans.  o;^-^,  or    '(li^OO.) 

4.  203'— ;?:2=:1.  Ans.  a:i=:l,  or  ^(— l=ty_7). 

5.  .^3_3^2_|_^_^2=0.  Ans.  .T=2,  or  ^(l±,/5). 

6.  .T'=6.r4-9.  Ans.  .T:^3,   or  ^(— 3±|/=^). 

7.  .-^+70:^=22.  Ans.  x=S,  or  29±7vA=T0. 

a:+7a:^— 22.=(ar— 8)  +  7(a;^— 2).     x^~2  is  a  divisor. 

8.  x'^\^r'—S9x=-Sl. 

Ans.  ar^=t3,  or  ^4(— 13±|/— 155). 

An  artifice  that  is  frequently  employed,  consists  in  adding  to 
each  side  of  the  equation,  such  a  number  or  quantity  as  will  render 
both  sides  perfect  squares. 


9. 


(jriven   X— ' — ^- — ,  to  find  x. 


x~b      ' 


Clearing  of  fractions,  X^—3x=.12-\^8]/x. 

Add  x-\-4:  to  each  side,  and  extract  the  square  root. 

From  which  we  easily  find  x=zd,  4,  or  ^( — 3dr j/— 7). 


QUADRATIC  EQUATIONS.  233 

10.  .-3=2^+l»^. 

X 

11.  l?^'_i_i^_49=9_|_?.     Add  -,  to  each  side. 

4      '    X-  X  x^ 

Ans.  a:=2,  — f,  or   i(— 3±|/93). 

Via? 

12.  a;*+i^_34a:=16.     Ans.  2r=r=±2,  —8,  or  -~h 

(17a:  \2 
—y-  I    to  each  side. 

13.  x4  1+g^^  j  -(3a:^+^)=Y0. 

Ans.  ar==3,  —3',  or   ^(— Irt^— 251). 

9 
Divide  by  x^,  and  add  j—^  to  each  side.     (See  Ex.  15,  Art.  242.) 

Multiply  by  2,  and  add  o7T  +  |g  ^o  each  side. 

It.    o^  .      841  ,  ,,     232       1    ,  ^ 
15.  27.^-- 3-+.-^-3-,+5. 


Ans.  x=2,  — y,  or  i(_2dri/— 266). 

841  1 

Multiply  both  sides  by  3,  transpose  — ^   and  —^,  ami  add  1    to 

each  side  to  complete  the  square. 


We  shall  now  present  a  few  solutions  giving  examples  of 
other  artifices. 

l-\-x^ 
16.   -zf-     ;-^=«  ;  to  find  X. 

{\-\-xy 

l-fa:»=^a(l-fa:)4=.a(l+4a:+6a:2-f4a:-H^), 
(1— a)(l-f  a;4)==4a(.T  f  a:-'5)4  Ga.T-'. 

2d  Bk.  20 


234  RAY'S  ALGEBRA,  SECOND  BOOK. 

Dividing  by  a;2,  (1— a)  ^  a:2-f  _  \=-4a/  a:+-  \+6a, 
2     1        4a  /      ,  1  \_   6a 

Complete    the    square,    and    find   the  value    of   x-\ — ,  which  is 


2a±:j/2(l+a)^  ^^^^  ^^^.^  2p^  ^^^  ^^  ^^^^^  ^^^  ^^^_,_^^2_i. 


17.  a:^+'/=/«,  and  3/^+2/— ^.a^  to  find  a:  and  y. 
From  1st  equation,  y—X  40  . 

a 

From  2d  equation,  y=xx+]/; 

.1+7        _rt_  iC+T/         a 

Therefore,  aj  4a  =a:x-fy,  and  — r-^  = : 

'  '  4a       x^y* 

[x-j-yy=-Aa^^  or  x-\-y—2a] 
But,  x^=y2a,  since  a:+?/— 2a, 

Therefore,  ic— ?/2,  and  y--^y=2a, 

Whence,  y=^—l±:^8a^T),  and  a::=^(4a+l=p j/8a-t-l). 

When  two  unknown  quantities  are  found  in  an  equation, 
in  the  form  of  x-{-i/  and  xi/,  it  is  generally  expedient  to 
put  their  sum  x-\-y=^Sy  and  their  product  x?/=p. 

18.  Given  (x+y)(x>/-hl)=--lSxy         (1), 

(x'-{-/Xxy-{-l)=z20Sxy  (2),  to  find  X  and  y. 

Let  x-{^y=s,  and  xy=p;  then, 

s(iHl)=18p,  (1),  and 

(s2— 2p)(p2_^l)=208p2  (2). 

From  the  square  of  (1),  take  (2),  and  after  dividing  bj  22>,  we 

have 

«2_|p2_|.i^58p  (3). 

But,    ....       2s(p-fl)=36p,  from(2), 
And   ....  2p=  2p. 

Adding, .    .    .     (s  fp+l)2:^96p, 


QUADRATIC  EQUATIONS.  235 

But,    ...     .  p  +  l=lf; 

ISp  

Therefore,  .     .     .     s=4|/6p — "^    ,  or  s- — 4sy'6p=— 18p; 

From  which,    .     .     s=3|/t)p,  or  yUp. 

But,    ....  p+l=4|/6p— s=3/^,  or  yBp. 

Whence, .     .     .        p^26zbv/675,  or  2d=|/3,  and 

s=±v/{6(26zfc|/675j},  or  d=i/{6(2d=  v/3)J. 

Having  x-\-y  and  xy,  the  values  of  a:  and  ?/  are  easily  found., 
(Art.  246);  two  of  the  values  are  :c=7rh4,/3^  ?/=2rfr/3. 

19.  2(x-^i/y+l={x^^i/)(xi/-^x'-{-f)         (1), 
a;+3/=3  (2). 

Ans.  x=2,  y=l. 


on    1   ,     3       .1   ,     N3  A  l-^2adtzyl2a-S 

20.   l-fa:3=of(l+ic)'.  Ans.  x= — ■ — ts-^ ^ • 

^  ^  Z(l — a) 


31.  4_1    L-2a-"=l. 

a:^       it  \  x 


Ans.  a:=]ll±i/l— 8«±\'2±2(1— 8a)^4-8a}. 

22.  a:;4-3'+^i/(a;4-y)4-^'^y^==85,  Ans.  x=^6,  or  1. 

^3'+(^+y)'+a^i'(«H-y)=97.  3/=l,  or  6. 

2c+a^  c 

Ans.  cc=:^ — — ?— ,,  or 


\a-\-b)d'         {a—h)d' 

24.  (.T'+l)(x2+l)(a:+l)=30x^  __ 

Ans.a:=A(3±]/5). 

25.  x'+/=35,  Ans.  x=^,  2,  or  l±i|/22; 
a;^-f/=13.  y=2,  3,  or  lq=i|/22. 


236  RAY'S  ALGEBRA,  SECOND  BOOK. 

26.  -q^^a,  Ans.  x=J  j  2aZ>c(a^-H>c-aj)_  |  ^ 

a^-l-y  \   (  (^ab-i-ac—bc){ab-^hc—ac)  j  ' 

^cyz^  _    j(  2ahc{ab'\-hc~ac)  | 

a)+2~   '  ^~  \  I  {ac-^bc—ab)(ab-\-ac—bc)  j  ' 

X1/Z /  f  2abc{ab-\-ac — be)  ) 

y-{-z       '  \  1  {ac-\-bc — ab)(ab-^bc — ac)  )  ' 

27.  (x^-^i)y=(f+r)c^, 

Ans.  x= 


i{^r3+3+^>3-: 


y=i{r3.^/r3+3±^V9-l} 


VIII.   RATIO,   PROPORTION",  AND 
PROGRESSIONS. 

SS4.  Two  quantities  of  the  same  kind  may  be  com- 
pared in  two  ways.     By  considering, 

1st.  How  much  the  one  exceeds  the  other. 

2d.    How  many  times  the  one  is  contained  in  the  other. 

The  first  method  is  termed  comparison  by  difference;  the 
second,  comparison  by  quotient.  The  first  is  sometimes  called 
Arithmetical  ratio;  the  second.  Geometrical  ratio. 

If  we  compare  2  and  6,  we  find  that  2  is  four  less  than 
0,  or  that  2  is  contained  in  6  three  times.  Also,  the  arith- 
metical ratio  of  a  to  b  is  b — a;  the  geometrical  ratio  of 

a  to  6  IS  -. 
a 

The  term  Ratio,  unless  it  is  otherwise  stated,  always 

signifies  g-sometrical  ratio. 


RATIO  AND  PROPORTION.  237 

255m  Ratio  is  the  quotient  which  arises  from  dividing 
one  quantity  by  another  of  the  same  kind.  Thus,  the 
ratio  of  2  to  6  is  3,  and  the  ratio  of  a  to  ma  is  m. 

S56.  When  two  numbers,  as  2  and  6,  are  compared, 
the  first  is  called  the  antecedent,  and  the  second  the  conse- 
quent. When  spoken  of  as  one,  they  are  called  a  couplet. 
When  spoken  of  as  two,  they  are  called  the  terms  of  the 
ratio. 

Thus,  2  and  6  together  form  a  couplet,  of  which  2  is  the 
first  term,  and  6  the  secojid  term. 

257.  Ratio  is  expressed  in  two  ways  : 

1st.  In  the  form  of  a  fraction,  of  which  the  antecedent 
is  the  denominator,  and  the  consequent  the  numerator. 
Thus,  the  ratio  of  2  to  6  is  expressed  by  £ ;  the  ratio  of 

a  to  6,   by  -. 

2d.  By  placing  two  points  between  the  terms.  Thus, 
the  ratio  of  2  to  6,  is  written  2:6;  the  ratio  of  a  to  h, 
a\h,  etc. 

258.  The  ratio  of  two  quantities  may  be  either  a  whole 
number,  a  common  fraction,  or  an  interminate  decimal. 

Thus,  the  ratio  of  2  to  6  is  fi,  or  3,  of  10  to  4,  is  |. 

The  ratio  of  2  to  ]/5  is  \-;  or  ^'^         ,  or  1.118-f . 
2  2 

2SO.  Since  the  ratio  of  two  numbers  is  expressed  by 
a  fraction,  of  which  the  antecedent  is  the  denominator,  and 
the  consequent  the  numerator,  whatever  is  true  with  regard 
to  a  fraction  is  true  with  regard  to  the  terms  of  a  ratio. 
Hence, 

1st.  To  multiply  the  consequent,  or  divide  the  antecedent 
of  a  ratio  hy  any  number,  multiplies  the  ratio  hy  that 
number. 


238  RAY'S  ALGEBRA,  SECOND  BOOK. 

2d.  To  divide  the  consequent  or  to  multiply  the  antecedent 
of  a  ratio  hy  any  number^  divides  the  ratio  by  that  number. 

3d.  To  multiply  or  divide  both  the  antecedent  and  conse- 
quent of  a  ratio  by  any  number,  does  not  alter  the  ratio. 

!3GO.  When  the  terms  of  a  ratio  are  equal  to  each 
other,  the  ratio  is  said  to  be  a  ratio  of  equality;  when  the 
second  term  is  greater  than  the  first,  a  ratio  of  greater 
inequality ;  when  it  is  less,  a  ratio  of  less  inequality. 

Thus,  the  ratio  of  2  to  2  is  a  ratio  of  equality. 

The  ratio  of  2  to  3  is  a  ratio  of  greater  inequality. 

The  ratio  of  3  to  2  is  a  ratio  of  less  inequality. 

Hence,  a  ratio  of  equality  may  be  expressed  by  1;  a  ratio  of 
greater  inequality,  by  a  number  greater  than  1 ;  and  a  ratio  of  less 
inequality,  by  a  number  less  than  1. 

S61.  When  the  corresponding  terms  of  two  or  more 
ratios  are  multiplied  together,  the  ratios  are  said  to  be 
compounded,  and  the  result  is  termed  a  compound  ratio. 

Thus,  the  ratio  of  a  to  6,  compounded  with  the  ratio  of  c  to  d,  is 
6      dbd 
a      c~aG' 

A  ratio  compounded  of  two  equal  ratios  is  called  a  dupli- 
cate ratio;  one  compounded  of  three  equal  ratios,  a  tripli- 
cate ratio. 

Thus,  the  duplicate  ratio  of  —  is  —  X  — =  — oj  the  triplicate  ratio 

G/  Ct        CL        CL 

.     63 

IS    -  -A. 

The  ratio  of  the  square  roots  of  two  quantities  is  called  a 
subduplicate  ratio;  that  of  the  cube  roots,  a  subtriplicate 
ratio. 


Thus,  the  subduplicate  ratio  of  4  to  9  is  |;  and  that  of  a  to  6  is 

1-:^;  the  subtriplicate  ratio  of  a  to  6  is  ^-^-. 
V«  fa 


RATIO  AND  PROPORTION.  239 

2^2*  Ratios  may  be  compared  with  each  other  by  re- 
ducing the  fractions  which  represent  them  to  a  common 
denominator. 

Thus,  the  ratio  of  2  to  7  is  greater  than  the  ratio  of  3  to  10,  for 
the  fractions  |  and  LO,  reduced  to  a  common  denominator,  are  y 
and  2J)^  and  the  first  is  greater  than  the  second. 


PROPORTION. 

S63.  Proportion  is  an  equality  of  ratios ;  that  is,  when 
two  ratios  are  equal,  their  terms  are  said  to  be  proportional. 

Thus,  if  the  ratio  of  a  to  6  is  equal  to  the  ratio  of  C  to  d ;  that 

b      cl 

is,  if  —  =  — ;  then,  a,  6,  C,  «,  form  a  proportion. 

Proportion  is  written  in  two  ways  : 

1st.  By  placing  a  double  colon  between  the  ratios ; 

Thus,  a:  b  .  :  c  :  d. 
Read,  a  is  to  6  as  c  is  to  d. 

2d.  By  placing  the  sign  of  equality  between  the  ratios ; 

Thus,  a  :  6=C  :  d. 
Read,  the  ratio  of  a  to  6  equals  the  ratio  of  c  to  d. 

From  the  preceding  definition  it  follows,  that  when  four 
quantities  are  in  proportion,  the  second  divided  by  the 
first,  must  give  the  same  quotient  as  the  fourth  divided  by 
the  third.  This  is  the  primary  test  of  the  proportionality 
of  four  quantities. 

Thus,  if  3,  5,  6,  10,  are  the  four  terms  of  a  proportion,  so  that 
3  :  5  :  :  6  :  10,  we  must  have  l^^^'zP. 

If  these  fractions  are  not  equal  to  each  other,  the  pro- 
portion is  false. 

Thus,  the  proportion  3  :  8  ;  :  2  :  5  is  false,  since  |>|. 


240  RAY  S  ALGEBRA,  SECOND  BOOK. 

Remark. — The  words  ratio  and  proportion  should  not  be  con- 
founded. Thus,  two  quantities  are  not  in  the  proportion  of  2  to  3, 
but  in  the  ratio  of  2  to  3.  A  ratio  subsists  between  two  quantities, 
a  proportion  between  four. 

264.  Each  of  the  four  quantities  in  a  proportion  is 
called  a  term.  The  first  and  last  terms  are  called  the  ex- 
tremes; the  second  and  third  terms,  the  means. 

S6«>.  Of  four  quantities  in  proportion,  the  first  and 
third  are  called  the  antecedents^  and  the  second  and  fourth, 
the  consequents  (Art.  257)  ;  and  the  last  is  said  to  be  a 
fourth  proportional  to  the  other  three  taken  in  their  order. 

!366.  Three  quantities  are  in  proportion  when  the  first 
has  the  same  ratio  to  the  second,  that  the  second  has  to  the 
third.  The  middle  term  is  a  mean  proportional  between 
the  other  two. 

Thus,  if a:  b  :  :  b:  c, 

then  6  is  a  mean  proportional  between  a  and  c;   and  C  is  called  a 
third  proportional  to  a  and  b. 

When  several  quantities  have  the  same  ratio  between  each 
two  that  are  consecutive,  they  are  said  to  form  a  continued 
proportion. 

207.  Proposition  I. — In  every  proportion^  the  product 
of  the  means  is  equal  to  the  product  of  the  extremes. 

Let a  \  b  :  \  c  :  d. 

Since  this  is  a  true  proportion,  we  must  have  (Art.  263) 

a~  c' 

Clearing  of  fractions,  bc^=.ad. 

Illustration  by  numbers.     2  :  6  :  5  :  15;  and  6x5=2x15. 

,.    ,    ,    6c        ad  ,     ad        be    „ 
Taking  bc—ad,  we  tnd  d= — ,  c=-^,  o= — ,  «=^-    Hence, 


RATIO  AND  PROPORTION.  24 1 

If  any  three  terms  of  a  proportion  he  given,  the  remaining 
term  may  he  found. 

1.  The  first  three  terms  of  a  proportion  are  cc-f  ?/,  a-} — 3/^, 
and  X — y;  what  is  the  fourth?  Ans.  x^ — Ixy-^-y"^. 

2.  The  1st,  3d,  and  4th  terms  of  a  proportion  are  (ni — ?i)^, 
m^ — ?i^,  and  m-\-n  ;  required  the  2d.  Ans.  m — n. 

3.  The   1st,    2d,    and    4th    terms   of   a    proportion    are 

^^±4,  a^_6,   and  (^=VM^Z^  •  required  the  3d. 
a— 1/6  a-\-yb 

Ans.  1. 

This  proposition  furnishes  a  more  convenient  test  of  proportion- 
ality than  the  method  given  in  Art.  263. 

Thus,  2  :  3  :  :  5  :  8,  is  not  a  true  proportion,  since  3X5  is  not 
equal  to  2x8. 

!S68.  Proposition  II. — Conversely,  If  the  jyroduct  of 
two  quantities  is  equal  to  the  product  of  two  others,  tvfo  of 
them  may  be  made  the  means,  and  the  other  two  the  extremes 
of  a  proportion. 

Let bcz=ad. 

Dividing  each  of  these  equals  by  ac,  we  have 

be     ad 

OG     ac' 

b      d 

Or, -  =  -. 

'  a     c 

That  is  (Art.  263),    .     .     .     a  :  b  :  .  c  :  d. 

By  dividing  each  of  the  equals  by  ab,  cd,  bd,  etc.,  we  may  have 
the  proportion  in  other  forms. 

Or,  since  one  member  of  the  equation  must  form  the  extremes  and 
the  other  the  means,  we  have  the  following 

Rule. —  Take  either  factor  on  either  side  of  the  equation 
for  the  first  term  of  the  proportion,  the  two  on  the  other  side 
for  the  second  and  third,  and  the  remaining  factor  for  the 
fourth. 

2d  Bk.  21* 


242 


RAY'S  ALGEBRA,  SECOND  BOOK. 


Thus,  from  each  of  the  equations  bc::=zCid,  and  3x12=^4x9,  we 
may  have  the  eight  following  forms  : 


a:  b: 

:c:d. 

3 

4 

•  :    9: 

12. 

a:  C: 

:b:d. 

3: 

9 

:    4: 

12. 

d:b: 

:  c:  a. 

12 

4 

:    9: 

3. 

d:c: 

:b.a. 

12 

9 

:    4: 

3. 

b:a: 

:  d:  C. 

4: 

3 

:  12: 

9. 

b.d: 

:  a:  C. 

4: 

12 

:    3: 

9. 

C:  a  : 

:d:b. 

9: 

3 

:  12: 

4. 

c:d: 

:  a:  b. 

9: 

12 

:    3: 

4. 

SOO.  Proposition  III. — If  three  quantities  are  in  pro- 
portion, the  product  of  the  extremes  is  equal  to  the  square 
of  the  mean. 

If a:  b::  b:  C] 

Then,  (Art.  267),  .     .     .     ac=^bb^b^. 

It  follows  from  Art.  268,  that  the  converse  of  this  proposition  is 
also  true. 

Thus,  if ac=62, 

a  :  b  :  :  b  :  c.     Hence, 

If  the  product  of  the  first  and  third  of  three  quantities  is 
equal  to  the  square  of  the  second.,  the  second  is  a  mean  pro- 
portional between  the  first  and  third. 

STO.  Proposition  IV. — If  four  quantities  are  in  pro- 
portion, they  will  he  in  proj>ortion  hij  ALTERNATION;  that 
is,  the  first  will  he  to  the  third  as  the  second  to  the  fourth. 


Let 

Then,  (Art.  263),      .     . 

Multiply  both  sides  by  c. 

Divide  both  sides  by  b, 

That  is,  (Art.  263),     .     . 
If 2:6:: 


d; 


.     a  :  b  :  :  C  :  d. 
b      d 
a~  g' 
be 
a 

c  _d 
a~b' 

a  :  C:  :  b  :  d. 
10:30;  then,  2  :10  :  :6:30. 


RATIO  AND  PROPORTION.  243 

S71.  Proposition  V. — If  four  quantities  are  in  propor- 
tion, they  will  he  in  proportion  hy  Inversion  ;  that  is,  the 
second  will  he  to  the  first  as  the  fourth  to  the  third. 

Let a  .  b  :  :  C  :d. 

Then,  (Art.  263),      .     . 


a      G  ' 

T         .       ,  •  a     c 

Inverting  the  fractions,    .     .    j^z=z-^. 

That  i«,  (Art.  263),       .     ,     b  :  a  :  :  d  :  C. 

If 5  :  10  :  :  6  :  12;   then,  10  :  5  :  :  12  :  6. 

6      d 

It  follows  from  this  proposition,  that  the  equation  —  =  —  may  be 

converted  into  a  proportion  in  either  of  two  ways,  thus  : 
a  :  b  :  :  c  :  d,  or  b  •  a  :  \  d  :  G. 


1S7^*  Proposition  VI. — If  two  sets  of  proportions  have 
an  antecedent  and  consequent  in  the  one,  equal  to  an  ante- 
cedent and  consequent  in  the  other,  the  remaining  terms  will 
he  proportional. 

Let a  .  b  .  :  G:  d        (1), 

And a.b:  .e.f  (2); 

Then  will    .     .     .     .     G  :  d  :     e    f. 

From  (1),  -  =  -;  from  (2),  ~^'-.     Hence,  ~=*  \ 

Which  gives    .     .     .     G  .  d  :  :  e  :  f. 

If  4  :  8  :  :  10  :  20  and  4  ;  8  :  :  6  :  12;  then,  10  :  20  :  :  6  :  12. 


273.  Proposition  VII. — If  four  quantities  are  in  pro- 
-portion,  they  will  he  in  proportion  hy  COMPOSITION  ;  that 
is,  the  sum  of  the  first  and  second  will  he  to  the  first  or 
'second,  as  the  sum  of  the  third  and  fourth  is  to  the  third  or 
fourth. 


244  RAY'S  ALGEBRA,  SECOND  BOOK. 

Let a\b'.\c:d        (1), 

Then  will a+6  :  6  :  :  c+d  :  d. 

^        ,-,.  b      d 

From  (1) -  =  -. 

,     .       •  (^     c 

Inverting  the  fractions,    .     t^  =  ^- 

Adding  unity  to  both  members,    r+l=;y+l- 

^   ,     •  .  ,.       •         a+6     cA-d 

Reducing  to  improper  tractions,  — ^- — =:     '     . 

Hence,  (Art.  271),    .     .   a^b  :  b  :  :  c-^d  :  d. 

If  3  :  6  :  :  9  :  18;   then,  3+6  :  6  :  :  9+18  :  18,  or  9  :  6  :  :  27  :  18. 

In  a  similar  manner  it  may  be  shown  that  a-\-b  :  a  :  :  c-{-d  :  c. 


S74:.  Proposition  VIII. — If  four  quantities  are  in  pro- 
portion, they  will  he  in  proportion  hy  Division  ;  that  is,  the 
difference  of  the  first  and  second  will  he  to  the  first  or  sec- 
ond, as  the  difference  of  the  third  and  fourth  is  to  the  third 
or  fourth. 

Let a:  b:\  C:  d         (1), 

Then  will a—b  :  b  :  :  (i—d  :  d. 

b      d 

From  (1), -  =  -. 

,     .       .  «     c 

Inverting  the  fractions,    .     r=  j- 

Ct  G 

Subtracting  unity  from  both  members,  j- — 1=^ — 1. 

T.   ,     .  .  .       .  <^—h     G—d 

Reducing  to  improper  fractions,    .     .     — v— = — -r— • 

This  gives  (Art.  271),      a~b  :  b  :  :  G—d :  d. 

If  18  .  6  .  :  30  :  10;  then,  18-6  :6  ;  :  30-10  :  10,  or  12  :  6  ;  20  :  10. 

In  a  similar  manner  it  may  be  shown  that  a — 6 :  a  :  :  c — d :  c. 


S7S«  Proposition  IX. — If  four  quantities  are  in  pro- 
portion, the  sum  of  the  first  and  second  will  he  to  their 
difference  as  the  sum  of  the  third  and  fourth  is  to  their  dif- 
ference. 


RATIO  AND  PROPORTION.  245 

Let a:  d  :  :  c:  d         (1), 

Then  will    ....  a-f6  :  a—b  :  :  e-f  d  :  c — d. 

From  (1),  by  Composition  and  Division,  (Arts.  273,  274,) 
a-f6  :  6  :  :  C-f  C?  :  d; 

And a—b  :  b  :  :  c — d  :  d. 

By  alternation,   .     .    a-f  6  :  C-\-d  :  :  b  :  d; 

And a — b  :  C—d  :  :  b  :  d. 

From  which,  (Art.  272),  a+6  :  C-^d  :  :  a—b  :  c—d. 
Or,  by  alternation,  a+6  :  a— 6  :  :  c-\-d  :  c— d. 

If  6  : 2  :  :  12  :  3  ;  then,  6+2  :  6  -2  :  :  12+4 :  12—4,  or  8 :  4  : :  16  :  8. 

!S76.  Proposition  X. — If  four  quantities  are  in  propor- 
tion^ like  i^owers  or  roots  of  those  quantities  will  also  he  in 
proportion. 

Let a.b  ::  C\  d, 

Then  will a"  :  6"  :  :  c"  :  d'*. 

From  the  1st, -==-.    Raising  each  of  these  equals 

6"      d" 
to  the  n<'' power,       .     .     •  ^.  =  -^»- 

That  is, a"  :  S** :  :  c"  :  d" 

Where  n  may  be  either  a  whole  number  or  a  fraction. 

If  2  :  6  :  :  10  :  30;  then,  22  :  62  :  :  102  :  302,  or  4  :  36  :  :  100  :  900. 

If  8:27:  :  64:  216;  then,  fS":  f27::  ^64:^216;  or  2:  3::  4:6. 

STT.  Proposition  XI. — If  two  sets  of  quantities  are  in 
proportion,  the  products  of  the  corresponding  terms  will  also 
he  in  proportion. 

Let a:b::C:d         (1), 

And m-.n-.'.r-.s         (2), 

Then  will am  :  bn  :  :  cr  :  ds. 

For  from  (1),    |  =  |  (3);    and  from  (2),  ^  =  ^  (4). 
Multiplying  (3)  by  (4)  —  = — ;  this  gives,  am  :  bn  :  :  cr  :  ds. 

(XTih        C/ 


Let     .     .     . 

.     . 

Then,     .     . 

. 

Since     a  :  h  : 

:  c 

Since  a  :  h  \ 

:  m 

246  RAY  S  ALGEBRA,  SECOND  BOOK. 

If      3  :  9  :  :  2  :  6,  and  5  :  15  :  :  4  :  12;  then,  15  :  135  :  :  8  :  72. 

278.  Proposition  XII. — Li  any  number  of  proportions 
having  the  same  ratio ^  any  antecedent  is  to  its  consequent  as 
the  sum  of  all  the  antecedents  is  to  the  sum  of  all  the  con- 
sequents. 

.     a  :  b  :  :  C:  d:  :  m:  n,  etc. 

.    a  :  b  :  :  a-\-c-^m  :  6-f rf-fn. 

d,  we  have     bc—ad  (Art.  267). 

w,  we  have  bm=^an, 
Also, ab=ab.    The  sum  of  these  equal- 
ities gives  ....    ab-\-bc-\-bm=^ab'\-ad^an. 
Factoring,      .    .    .    b{a-\-c-\-7n)=:za[b-\-d^n). 
This  gives  (Art.  268),  a  :  b  :  :  a-^C-\-m  :  b-\-d-{-n. 

If  5  :  10  :  :  2  :  4  :  3  :  6,  etc. ;  then,  5  :  10  :  :  5+2+3  :  10+4+6, 
or  5  :  10  :  :  10  :  ZO. 

EXERCISES  IN  RATIO  AND  PROPORTION. 

1.  Which  is  the  greater  ratio,  that  of  3  to  4,  or  3^  to 
4'^?  Ans.  last. 

2.  Compound  the  duplicate  ratio  of  2  to  3  ;  the  triplicate 
ratio  of  3  to  4 ;  and  the  subduplicate  ratio  of  64  to  36. 

Ans.  1  to  4. 

3.  What  quantity  must  be  added  to  each  of  the  terms 

of  the  ratio  m  :  n,  that  it  may  become  equal  to  p  :  q? 

,         mg — np 

Ans.  — -. 

p—q 

4.  If  the  ratio  of  a  to  h  is  2|,  what  is  the  ratio  of  2a 
to  h,  and  of  3a  to  4h?  Ans.  1],  and  3i}. 

5.  If  the  ratio  of  a  to  Z>  is  1^,  what  is  the  ratio  of  a-\-h 
to  b,  and  of  b — a  to  a?  Ans.  |,  and  |. 

6.  If  the  ratio  of  m  to  n  is  :j,  what  is  the  ratio  of  m — n 
to  6m,  and  also  to  5»?  Ans.  14,  and  6|. 


RATIO  AND  PROPORTION.  247 

7.  If  the  ratio  of  by — 8a;  to  Ix — by  is  6,  what  is  the 
ratio  of  a;  to  ^?  Ans.  7  to  11. 

8.  What  eight  proportions  are  deducible  from  the  equa- 
tion ah=^a^ — x^.  Ans.  a  :  a-j-rc  :  :  a — x  :  b, 

a  :  a — x  :  :  a-]-x  :  b, 

b  :  a-{-x  :  :  a — x  :  a,  etc. 

9.  If  x^-{-y^=2aXj  what  is  the  ratio  of  x  to  y? 

Ans.   X  :  y  :  :  y  :  2a — x. 

10.  Four  given  numbers  are  represented  by  a,  5,  c,  d ; 

what  quantity  added  to  each  will  make  them  proportionals  ? 

.  be — ad 

Ans. -j- 

a — b — c-\-d 

11.  If  four  numbers  are  proportionals,  show  th»t  there 
is  no  number  which  being  added  to  each,  will  leave  the 
resulting  four  numbers  proportionals. 

12.  Find  X  in  terms  of  y  from  the  proportions  x:y  \:a^  :b\ 
and  a  :  b  :  :  f^c-\-x  :  f^d-\-y. 

13.  Prove  that  equal  multiples  of  two  quantities  are 
to  each  other  as  the  quantities  themselves,  or  that 
ma  '.  mb  :  :  a  :  b. 

14.  Prove  that  like  parts  of  two  quantities  are  to  each 

other  as  the  quantities  themselves,  or  that  -  :  -  :  :  a  :  b. 

n    n 

15.  If  a  :  5  :  :  c  :  (7,  prove  that  ma  :  mb  :  \  nc  :  nd,  and 
also  that  ma  :  nb  :  :  mc  :  nd,  m  and  n  being  any  multiples. 

16.  Prove  that  the  quotients  of  the  corresponding  terms 
of  two  proportions  are  proportional. 

STd.  The  following  examples  are  intended  as  exercises 
in  application  of  the  principles  of  proportion. 

1.  Resolve  the  number  24  into  two  factors,  so  that  the 
sum  of  their  cubes  may  be  to  the  difference  of  their  cubes 
as  85  to  19. 


248 


RAY'S  ALGEBRA,  SECOND  BOOK. 


Let  X  and  y  denote  the  required  factors;  then,  xy=2A^  and 


X^-[-y^  .  x^ — y^ 
Therefore,  (Art.  275),  2x^  :  2y^ 

Or, x^  :    y^ 

Or,  (Art.  276),     ....        x   :    y 


19; 
IG: 


From  which  y=^X]  then,  substituting  the  value  of  y  in  the  equa- 
tion xy=24:,  we  find  x=zh6',  hence,  y=ziz4:. 


2.  Given  l^±l+l^^2,  to  find  x. 
if  x+  1— f/a;_l 

Resolving  this  equation  into  a  proportion,  we  have 


fx-\-l~fx—l  :  fx-{-l-\-f'X—l  :  :  1  :  2; 

.-.  (Art.  275),  2f  ^+1  :  2fx—l  :  :  3  :  1 ; 

Or, f  ^+1  :  ^^Hl  :  :  3  :  1 ; 

Or,  (Art.  276),     .     .     .     x-{-l  :  x—1  :  :  27  :  1 ; 

(Art.  275),     .     .     .     2a: :  2  :  :  28  :  26 ; 
Whence, 52a:=56,  or  a;=lJ^. 


ir 


}  : 


8.  x-\-y  :  x—y  :  :  3  :  1, 
x^ — ^3^=56. 

4.  x-\-y  :  a:  :  :  7  :  5, 1       .     . 
xy-{-y'=126.        j        .     . 

5.  (x-\-yy  :  (x~yy  :  :  64  :  1, 
a;y=63. 


6. 


—j/a'—x'' 


a-^y'd'—x' 


.     Ans.  a:=:4, 

Ans.  a;=±15, 

Ans.  .7;==t9, 

6+1  • 


.ns  x 


2ah 


7.  y/^j^-T/g-^^l ^^3  ^_ 

^a-\-x-\-^a — x      ^ 

8.  It  is  required  to  find  two  numbers  whose  product  is 
320,  and  the  diflference  of  whose  cubes  is  to  the  cube  of 


their  difference,  as  61  is  to  1. 


Ans.  20  and  16. 


RATIO  AND  PROPORTION.  249 

280.  Harmonical  Proportion. — Three  or  four  quan- 
tities are  in  Harmonical  Proportion  when  the  first  has  the 
same  ratio  to  the  last,  that  the  difference  between  the  first 
and  second  has  to  the  difi"erence  between  the  last  and  the 
last  except  one. 

Thus,  a,  6,  C,  are  in  harmonical  proportion  when  a  :  c  :  :  a — b  : 
b — c ;  and  a,  6,  c,  cZ,  when  a  :  d  :  :  a—b  :  c — d. 

1.  Let  it  be  required  to  find  a  third  harmonical  propor- 
tional X,  to  two  given  numbers  a  and  b. 

We  have, a  :  x  :  :  a — 6:5 — x\ 

Therefore,  (Art.  267),      a{b—x)^x{a—b)\ 

^^^^^^^^' ^^2£z6- 

2.  Find  a  third  harmonical  proportional  to  3  and  5. 

Ans,  15. 

3.  Find   a  fourth    harmonical   proportional   ic,  to  three 

c;iven  numbers,  a,  h,  and  c.  ^  ac 

^  Ans.  x= 


2a— b' 

381.  Variation,  or,  as  it  is  sometimes  termed,  Gen- 
eral Proportion,  is  merely  an  abridged  form  of  common 
Proportion. 

Variable  Quantities  are  such  as  admit  of  various  values 
in  the  same  computation. 

Constant,  or  Invariable  Quantities  have  only  one  fixed 
value. 

One  quantity  is  said  to  var?/  directly  as  another,  when 
the  two  quantities  depend  upon  each  other  in  such  a  man- 
ner, that  if  one  be  changed  the  other  is  changed  in  the 
same  ratio. 

Thus,  the  length  of  a  shadow  varies  directly  as  the  height 
of  the  object  which  casts  it. 

Such    a   relation   between   A   and  B  is  expressed  thus, 


250  HAY'S  ALGEBRA,  SECOND  BOOK. 

A  oc  B,   the   symbol    oc    being   used  instead  of  varies^   or 
varies  as. 

282.  There  are  four  different  kinds  of  Variation,  which 
are  distinguished  as  follows : 

I.  A  X  B.  Here  A  is  said  to  vary  directly  as  B,  or, 
simply  A  varies  as  B. 

Ex. — If  a  man  works  for  a  certain  sum  per  day,  the 
amount  of  his  wages  varies  as  the  number  of  days  in  which 
he  works. 

II.  A  oc  ^.     Here  A  is  said  to  vary  inversely  as  B, 

Ex. — The  time  in  which  a  man  may  perform  a  journey 
will  vary  inversely  as  the  rate  of  traveling. 

III.  A  X  BC.  Here  A  is  said  to  vary  as  B  and  C 
jointly. 

Ex. — The  wages  to  be  received  by  a  workman  will  vary 
jointly  as  the  number  of  days  he  works,  and  the  wages 
per  day. 

T> 

IV.  A  X  ^.  Here  A  is  said  to  vary  directly  as  B,  and 
inversely  as  C. 

Ex. — The  time  occupied  in  a  journey  varies  directly  as 
the  distance,  and  inversely  as  the  rate  of  travel. 

These  four  kinds  of  variation  may  be  otherwise  modi- 
fied ;  thus,  A  may  vary  as  the  square  or  cube  of  B,  in- 
versely as  the  square  or  cube,  directly  as  the  square  and 
inversely  as  the  cube,  etc. 

Ex. — The  intensity  of  the  light  shed  by  any  luminous 
body  upon  an  object  will  vary  directly  as  the  size  of  the 
luminous  body,  and  inversely  as  the  square  of  its  distance 
from  the  object.     (See  Art.  238.) 


RATIO  AND  PROPORTION.  251 

In  the  followiBg  articles,  A,  B,  C,  represent  corresponding  values 
of  any  variable  quantities,  and  a,  6,  c,  any  other  corresponding 
values  of  the  same  quantities. 

SS3.  If  one  quantify  vary  as  a  second,  and  that  second 
as  a  third,  the  first  varies  as  the  third. 

Let  A  oc  B,  and  B  oc  C,  then  shall  A  oc  C.  For 
A  :  a  :  :  B  :  ft,  and  B  :  ^  :  :  C  :  c;  therefore,  (Art.  272), 
A  :  a  :  :  C  :  c ;  that  is,  A  oc  C. 

In  a  similar  manner  it  may  be  proved  that  if  A  oc  B, 

and   B  oc  ^,  that  A  oc  ^. 

284.  If  each  of  two  quantities  vary  as  a  third,  their  sum, 
or  their  difference,  or  the  square  root  of  their  j^'i'odiict,  will 
vary  as  the  third. 

Let  A  oc  C,  and  B  oc  C  ;  then,  A±B  oc  C;  also,  |/AB  oc  C. 

By  the  supposition,     .     .     .  A  :  a  :  :  C  :  e 

Therefore, A  :  a  :  :  B  :  b 

Alternately,  (Art.  270),  .     .  A  :  B  :  :  a  :  6 

By  Composition  or  Division,  AzfcB  :  B  :  :  adzb  :  6; 

Alternately, A±B  :  adzb  :  :  B  :  6  :  :  C  :  C; 

That  is, AdbB  x  C. 

Again, A  :  a  :  ;  C  :  C; 

And, B:6::C:e; 

Therefore,  (Art.  277),      .     .  AB  :  a6  :  :  C2  :  c^; 

And,  (Art.  276),     ....  ^/AB  :  ^ab~:  :  C  :  C; 

That  is, "i/AJToc  C. 

By  a  similar  method  of  reasoning,  the  following  propo- 
sitions may  be  proved: 

285.  If  one  qnantity  vary  as  another,  it  will  also  vary 
as  any  multiple,  or  any  part  of  the  other. 

That  is,  if  A  oc  B;  then,  A  oc  ?7iB,  or    oc  — . 


252  RAYS  ALGEBRA,  SECOND  BOOK. 

SSO*  If  one  quantity  vary  as  another,  any  power  or  root 
of  the  former  will  vary  as  the  same  power  or  root  of  the 
/after. 

Let  A  cc  B ;  then,  A"  cc  B",  n  being  integral  or  frac- 
tional. 

28T.  If  one  quantity  vary  as  another,  and  each  of  them 
be  multiplied  or  divided  by  any  quantify,  variable  or  invari- 
able, the  products  or  quotients  will  vary  as  each  other. 

A      B 
Let  A  oc  B  ;   then,  qA.  oc  qB,  and  —  oc  — . 

288.  If  one  quantity  vary  as  two  others  jointly,  either 
of  the  latter  varies  as  the  first  directly,  and  tlie  other  in- 
versely. 

A  A 

Let  A  Gc  BC ;    then,  B  oc  — ,  and   C  oc  y-- 

389.  If  A  vary  «s  B,  A  is  equal  to  B  multiplied  by 
some  constant  quantity. 

Let  A  oc  B ;  then,  A=mB. 

If  we  know  any  corresponding  values  of  A  and  B,  the 
constant  quantity  m  may  be  found. 

SOO.  In  general,  the  simplest  method  of  treating  varia- 
tions, is  to  convert  them  into  equations. 

1.  Given  that  y  oc  the  sum  of  two  quantities,  one  of 
which  varies  as  x,  and  the  other  as  x^,  to  find  the  corre- 
sponding equation. 

Because  one  part   oc  a:,  let  this  =zmx^ 
and  the  other  part    oc  x^,    "      "     =z,nx-. 

Therefore, y—mx-\-nx'^, 

where  m  and  n  are  two  unknown  invariable  quantities  which 
can  only  be  found  when  we  know  two  pairs  of  corresponding  values 
of  X  and  y. 


RATIO  AND  PROPORTION.  253 

2.  If  i/=r-{-s,  where  r  cc  x  and  s  a  — ,  and  if,  when  x=^l, 

X 

y=Q,  and  when  x=.2^  y^9>  what  is  the  equation  between 
X  and  yl 

n  n 

Let  r=7nx,  and  s=-  .-.  y—mxA — . 
'  X        ^  ^x 

But  if  a;=l,  ?/=6,  .-.  6=im+n; 

And  if  a;=2,  2/=9,  .-.  9=2m+^. 

2 
Hence,  w=4,  n— 2,  and  2/=4a:4- -. 

3.  If  y  cc  .T,  and  when  a;=2,  y=:^a ;  find  the  equation 
between  x  and  y.  Ans.  y=2ax. 

4.  If  3/  oc  -,  and  when   a;^^,  3/=8  ;    find  the  equation 

X  ^ 

between  x  and  y.  Ans.  y=-. 

X 

5.  If  3/=  the  sum  of  two  quantities,  one  of  which  varies 

as  X.  and  the  other  varies  inversely  as  x^ ;  and  when  x=.\, 

y=6,  and  when  x=^2,  y=^  ',    find  the   equation  between 

X  and  y.  ^4 

Ans.  y=2x-\--. 

6.  Given  that  y=  the  sum  of  three  quantities,  of  which 
the  1st  is  invariable,  the  2d  varies  as  x,  and  the  3d  varies 
as  x"^.  Also,  when  x=l,  2,  3,  y=Q,  11,  18,  respectively; 
find  y  in  terms  of  ic.  Ans.  y=S~{-2x-\-x^. 

7.  Given  that  s  gc  f,  when  /  is  constant;  and  s  oc/, 
when  t  is  constant;  also,  2s=/j  when  ^=1.  Find  the 
equation  between  /,  s,  and  t.  Ans.  s=-^ft\ 

Remarks.— 1.  The  above  examples  may  all  be  proved.  Thus, 
if  in  Ex.  5,  we  put  x=:^l  in  the  answer,  y  will  equal  6.  If  we  put 
x=2,  y=5. 

2.  The  Principles  of  Variation  are  extensively  applied  in  mechan- 
ical philosophy. 


254  RAY'S  ALGEBKA,  SECOND  BOOK. 


ARITHMETICAL    PROGRESSION. 

291.  An  Arithmetical  Progression  is  a  series  of  quan- 
tities whicli  increase  or  decrease  by  a  common  difference. 

Thus,  1,  3,  5,  7,  9,  etc.,  or  12,  9,  6,  3,  etc.,  and  a^ 
a-f-cZ,  a-j-2c?,  etc.,  a,  a — (7,  a — 2(Z,  etc.,  are  in  Arithmeti- 
cal Progression. 

The  series  is  said  to  be  increasing  or  decreasing^  accord- 
ing as  d  is  positive  or  negative. 

293.  To  investigate  a  rule  for  finding  any  terjn  of  an 
arithmetical  progression,  take  the  following  series,  in  which 
the  first  line  denotes  the  number  of  each  term,  the  second 
an  increasing  arithmetical  series,  and  the  third  a  decreas- 
ing arithmetical  series. 

12  3  4  5 

a,        a~^d,        a-f2c?,        a~^3d,        a-|-4c?,   etc., 
a,        a~d,        a— 2d,        aSd,        a~4d,   etc. 

It  is  manifest  that  the  coefficient  of  d  in  any  term  is 
less  by  unitT/  than  the  number  of  that  term  in  the  series ; 
therefore,  the  n^^  term  r=a-f-(n — l)d. 

If  we  designate  the  n^^  term  by  /,  we  have 

l^=a-{-{n — l)d,  when  the  series  is  increasing,  and 
l=a — (n — l)d,  when  the  series  is  decreasing.     Hence, 

Rule  for  finding  Any  Term  of  an  Arithmetical  Series. — 

Multij)ly  the  common  difference  by  the  number  of  terms  less 
one;  when  the  series  is  increasing,  add  this  product  to  the 
first  term;  when  decreasing,  subtract  it  from  the  first  term. 

The  equation  l  =  a-\-(ii — l)c7,  contains  four  variable 
quantities,  any  one  of  whicli  may  be  found  when  the  other 
three  are  known. 


ARITHMETICAL  PROGRESSION.  255 

S93.  Having  given  the  first  term  a,  the  common  dif- 
ference d^  and  the  number  of  terms  n,  to  find  S,  the  sum 
of  the  series. 

If  we  take  any  arithmetical  series,  as  the  following,  and  write 
the  same  series  under  it  in  an  inverted  order,  we  hLve 

S=  1+3  +  5+  7+  9+11, 
S^ll+9  +  7+  5+  3+  1. 


Adding,     .     .    28=12+12+12+12+12+12. 

2S=12X  the  number  of  terms,  =12x6=72. 
Whence,    .     .       S=:2  of  72=^36,  the  sum  of  the  series. 

To  render  this  method  general,  let  ^=  the  last  term,  and  write  the 
scries  both  in  a  direct  and  inverted  order. 

Then,     S=a+(a+d)+(a+2(^)+(a+3d).   .    .    +^, 
And,      S=;  +  (l—d)  +  {l—2d)  +  (l—Scl).  .    .     +a. 

2S=(H«)+(^+«)+(^+«)+(^+a).  .    .    +(Ha), 
2S=(^+a)  taken  as  many  times  as  there  are  terms  (r?)  in 
the  series. 
Hence,     ....    2S=(^+a)n; 

S=(Z+a)^=(^)n.    Hence, 

Rule  for  finding  the  Sum  of  an  Arithmetical  Series,— 
Multiply  half  the  sum  of  the  two  extremes  hy  the  number  of 
terms. 

It  also  appears  that 

The  sum  of  the  extremes  is  equal  to  the  siim  of  any  other 
two  terms  equally  distant  from  the  extremes. 

294.  The  equations  l=a^{n—\)d,  and  Sr^(a-^?)^, 
furnish  the  means  of  solving  this  general  problem : 

Knowing  any  three  of  the  five  quantities^  o,  d,  I,  n,  S, 
which  enter  into  an  arithmetical  series^  to  determine  the  other 
two. 

The  following  table  contains  the  results  of  the  solution  of  all  the 
diflFerent  cases.     As,  however,  it  is  not  possible  to  retain  these  in 


256 


RAY'S  ALGEBRA,  SECOND  BOOK. 


the  memory,  it  is  best,  in  ordinary  cases,  to  solve  all  examples  in 
Arithmetical  Progression  by  the  above  two  formulae: 


No. 

Given. 

Required. 

Formulae. 

1. 

2. 
3. 

4. 

a,  d,  n 
a,  cZ,  S 

a,  n,   S 
d,  n,   S 

l=a-\-{n—l)d, 
l=-ld-^^[2d^J^{a-ldY], 

^= a, 

n^       2 

5. 
6. 

7. 
8. 

a,  cZ,  n 
a,  rf,  I 

a,  n,  I 
d,  n,  I 

s 

S=An{2a+(7i— l)d}, 
^~   2     '      2d   ' 

%=\n\ll—[n—\)d\. 

9. 
10. 
11. 
12. 

a,  71,  I 
a,  w,   S 
a,  ?,    S 
n,  I,    S 

cZ 

^    2(S-a/i) 
n(n_l) ' 

'^-2S_^-a' 
2(n^-S) 
n(n-l)- 

13. 
14. 
15. 
IG 

a,  d,  I 

a,  d,  S 
a,  I,    S 
d,  I    S 

n 

-ty^  (2a— cZ)2-|-8dS— 2a+rf 
""-                      2d 
2S 

2^f-c^=fc/(2^+d)2-8c^s 

^                       2rf 

17. 
18. 
19. 
20. 

C/,    71,   I 

d,  71,  S 
d,  I,  S 
w,  I,   S 

a 

a=l—[n—\)d, 

S      (rt— l)c« 

a-- ^^ — o — J 

n           2 

a  _-??-.. 
n 

ARITHMETICAL  PROGRESSION.  257 

1.  Find  the  15"*  term  of  tlie  series  3,  7,  11,  etc. 

Ans.  59. 

Here,  a=3,  n — 1=14,  and  d=zA.  Substituting  these  values  in 
formula  (1),  we  have  ^=3+14x4=3-f  o6==59. 

2.  Find  the  20"*  term  of  the  series  5,  1,  —3,  etc. 

Ans.  —71. 

3.  Find  the  S''^  term  of  the  series  f ,  f^,  -^,  etc. 

Ans.  -^^ 

4.  Find  the  30"^  term  of  the  series  —27,  —20,  —13, 
etc.  Ans.  176. 

5.  Find  the  n'^  term  of  1  +  3+5-f  7.     Ans.  2n— 1. 

Of  2-f 2H2^  + Ans.  J(n-f  5). 

Of  13+122  +  12^  +  .     .     .     .  Ans.  K^O—Ji). 

6.  Find  the  sum  of  1-f  2-|-  3-|-4,  etc.,  to  50  terms. 

From  formula  (1),  we  find  ?=:50.  Substituting  this  in  formula 
(2),  we  Lave  S=(l -[-50)25=1275,  Ans.     Or,  use  formula  5. 

7.  Of  7+V+ 2^+,  etc.,  to  16  terms.         Ans.  142. 

8.  Of  12+8+4+,  etc.,  to  20  terms.       Ans.  —520. 

9.  Of  2+2i+2j+,etc.,  t0  7iterms.    Ans.^vC^+ll). 

10.  Of  ^— 2_y_,  etc.,  to  n  terms.    A.  4s(13— 7vO- 

11     f\Q  '^~~^    .  ^*~2      n — 3  ,        ^       ^         ^ 

11.  Ul 1 ^,  etc.,  to  n  terms. 

n  n  n  ^_^ 

Ans.  —^. 

12.  If  a  falling  body  descends  16j':7  ^"set  the  1st  sec, 
3  times  this  distance  the  next,  5  times  the  next,  and  so  on, 
how  far  will  it  fall  the  30th  sec,  and  how  far  altogether  in 
half  a  min.  ?  Ans.  948ii,  and  14475  ft. 

13.  Two  hundred  stones  being  placed  on  the  ground  in 
a  straight  line,  at  the  distance  of  2  feet  from  each  other; 

2d  Bk.  22 


258  RAY'S  ALGEBRA,  SECOND  BOOK. 

how  far  will  a  person  travel  who  shall  bring  them  sepa- 
rately to  a  basket,  which  is  placed  20  yards  from  the  first 
stone,  if  he  starts  from  the  spot  where  the  basket  stands  ? 
Ans.  19  miles,  4  fur.,  640  ft. 

14.  Insert  3  arithmetical  means  between  2  and  14. 

Here,  a=:2,  ^=14,  and  n^5.  From  formula  (1),  we  obtain  d=S. 
Hence,  the  three  means  will  be  5,  8,  and  11. 

To  solve  this  problem  generally,  let  it  be  reqiiired  to  insert  m 
arithmetical  means  between  a  and  I. 

Since  there  are  m  terms  between  a  and  I,  we  shall  have  n=m-\-2, 

and    formula    (1)    becomes    I  ^:^  a -\-  (7n4-l)d.      Hence,    d^=- -. 

^    ^  I    V         I      ;  J  7/1+1 

Therefore, 

77ie  common  difference  will  he  equal  to  the  difference  of  the 
extremes  divided  hy  the  number  of  means  plus  one. 

15.  Insert  4  arithmetical  means  between  3  and  18. 

Ans.  6,  9,  12,  15. 

16.  Insert  9  arithmetical  means  between  1  and  — 1. 

Ans.  I,  I,  etc.,  to  —4. 

17.  How  many  terms  of  the  series  19,  l7,  15,  etc., 
amount  to  91?  Ans.  13,  or  7. 

From  (2)  and  (1),  find  n,  or  use  formula  14.  Explain 
this  result. 

18.  How  many  terms  of  the  series  .034,  .0344,  .0348, 
etc.,  amount  to  2.748?  Ans.  60. 

19.  The  sum  of  the  first  two  terms  of  an  arithmetical 
progression  is  4,  and  the  fifth  term  is  9  ;  find  the  series. 

Ans.  1,  3,  5,  7,  9,  etc. 

20.  The  first  two  terms  of  an  arithmetical  progression 
being  together  =18,  and  the  next  three  terms  =12,  how 
many  terms  must  be  taken  to  make  28?       Ans.  4,  or  7. 

21.  In  the  series  1,  3,  5,  etc.,  the  sum  of  2r  terms: 
the  sum  of  r  terms  :  :  x  :  1  ;  determine  the  value  of  x. 

Ans.  4. 


GEOMETRICAL  PROGRESSION.  259 

22.  A  sets  out  for  a  certain  place,  and  travels  1  mile  the 
first  day,  2  the  second,  and  so  on.  Five  days  afterward 
B  sets  out,  and  travels  12  miles  a  day.  How  long  and 
how  far  must  B  travel  to  overtake  A? 

Ans.  3  days,  or  10  days;  and  travel  36  miles, 
or  120  miles.     Explain  these  results. 


GEOMETRICAL    PROGRESSION. 

S95.  A  Geometrical  Progression  is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding,  by  multiply- 
ing it  by  a  constant  quantity,  termed  the  ratio. 

Thus,  1,  2,  4,  8,  16,  etc.,  is  an  increasing  geometrical 
progression,  whose  common  ratio  is  2. 

Also,  54,  18,  6,  2,  etc.,  is  a  decreasing  geometrical  pro- 
gression, whose  common  ratio  is  \. 

In  general,  a,  ar^  ar'^,  ai^^  etc.,  is  a  geometrical  progres- 
sion, whose  common  ratio  is  r,  and  which  is  an  increasing 
series  when  r  is  greater  than  1 ;  but  a  decreasing  series 
when  r  is  less  than  1.     It  is  evident  that 

In  any  given  geometrical  series,  the  common  ratio  will  he 
found  hy  dividing  any  term  by  the  term  next  preceding. 

I296.  To  find  the  last  term  of  a  geometrical  progression. 

Let  a  denote  the  first  term,  r  the  common  ratio,  I  the 
n'^  term,  and  S  the  sum  of  n  terms ;  then  the  respective 
terms  of  the  series  will  be 

1,        2,        3,        4,        5,   .     .     .  n— 3,     n—2,    n— 1,      n, 
a,      ar,     ar^^    ar^^    ar*  .    .    .  ar"-'*,  ar"-^^  ar"-"^^  ar"-^. 

That  is,  the  exponent  of  r,  in  the  second  term,  is  1,  in 
the  third  term  2,  in  the  fourth  term  3,  and  so  on.  Hence, 
the  n}^^  term  of  the  series  will  be  l=^ar''~^.     Hence, 


260  RAY'S  ALGEBRA,  SECOND  BOOK. 

Rule  for  finding  the  Last  Term  of  a  Geometrical 
Series. — Multiply  the  first  term  hy  the  ratio  raised  to  a  power 
whose  exponent  is  one  less  than  the  number  of  terms. 

Required  to  find  the  6'^  term  of  the  geometrical  progres- 
sion whose  first  term  is  7,  and  common  ratio  2. 

2^=32;  and  7x32^=224,  the  6"^  term. 

SOT*  To  find  the  sum  of  all  the  terms  of  a  geometri- 
cal progression. 

If  we  take  the  series,  1,  3,  9,  27,  81,  and  represent  its 
sum  by  S;  then,  8=1-1-3  +  9  +  27  +  81  (a). 

Multiplying  by  the  ratio  3,  38=3+94-27+81+243         (6). 
Subtracting  (a)  from  (6),  3S— S=243— 1 ;  whence,  S=121. 

To  generalize  this  method,  let  a,  ar,  ar^,  ar^^  etc.,  be  any 
geometrical  series,  and  S  its  sum ;  then, 

S=a+ar+ar2+ar3.    .    .    .    +ar"-2+ar'»-i. 

Multiplying  this  equation  by  r,  we  have 

rS=ar+a7'2+ar3.    .    .    .     +ar"-i+ar'". 

Subtracting,    rS — S^ar** — a\  whence,  S=— '^ =— . 

Since, Izz^ar"*-^,  we  have  rl=^ar^\ 

n,,       «  o,    ar'^—Gi     rl — a     „ 

Therefore, S= ^  = =-.     Hence, 

'  r— 1         r— 1  ' 

Eule  for  finding  the  Sum  of  a  Geometrical  Series. — 
Multiply  the  last  term  hy  the  ratio^  from  the  product  subtract 
the  first  term,  and  divide  the  remainder  by  the  ratio  less  one. 

Find  the  sum  of  6  terms  of  the  progression  3,  12,  48,  etc. 
i=3x4^=3072.  ...  S=^  =  ?5!|^3^4095,  An. 

S9S.  If  the  ratio  r  is  less  than  1,  the  progression  is 
decreasing,  and  the  last  term  /,  or  ar''~^,  is  less  than  a.     In 


GEOMETRICAL  PROGRESSION.  261 

order  that  both  terms  oi  the  iraction  =■,  or =-   may 

r — 1'         r — 1         -^ 

be  positive,  change  the  signs  of  the  terms,  (Art.  124),  and 

S:=3 ,  or  =z-^ -.    Therefore,  for  finding  the  sum  of 

1 — r  1 — r 

the  series,  when  the  progression  is  decreasing, 


Kule. — Multiply  the  last  term  by  the  ratio,  subtract  the 
product  from  the  first  term,  and  divide  the  remainder  by  one 
minus  the  ratio. 

SOO.  When  the  series  is  decreasing,  and  the  number 
of  terms  infinite,   I  is  infinitely  small,    or   0.      Therefore, 

r/=::0,  and  S== becomes  8=^^ .     Hence, 

1 — r  1 — r 


Bale  for  finding  the  Sum  of  an  Infinite  Decreasing 
Series. — Divide  the  first  term  by  one  minus  the  ratio. 

Find  the  sum  of  the  infinite  series  l+^+l+gH-,  etc. 

Here,  a=l,   r=^,  and  S=:.j =  :^ 1^^^^'  ^°s. 

1     r      i      J 

That   the  sum   of  an  infinite  decreasing  series  may  bo 
finite,  will  easily  appear  from  the  following  illustration  : 

Take  a  straight  line,  AK,  and  bisect  it  in  B;  bisect  BK  in  C; 
CK  in  D,  and  so  on  continually;  then  will 

AK=zAB+BC+CD+,  etc.,  in  infinitum,   =AB+JAB-f  JAB,  etc., 
in  infinitum,  =:2AB,  which  agrees  with  the  example. 

300.  The  equations,  l=zar"-~^,  and  S=:- ^-,  furnish 

this  general  problem : 

Knowing  any  three  of  the  five  quantities  a,  r,  n,  I,  and  S, 
of  a  geometrical  progression,  to  determine  the  other  two. 


J^32 


RAY'S  ALGEBRA,  SECOND  BOOK. 


The    following  table  contains  all    the  values  of   each   unknown 
quantity,  or  the  equations  from  which  it  may  be  derived  : 


No. 


10. 

11. 

12 


13. 
14. 
15. 
16. 


Given. 


17. 

18. 
19. 
20. 


a,  r,  n 

a,  r,  S 

a,  n,  S 

r,  n,  S 


a,  r,  71 

a,  r,  I 

a,  n,  I 

r,  n,  I 


r,   n,  S 

r,   I,    S 
n,  I,    S 


a,  n,  I 

a,  w,  S 

a,  I,  S 

n,  I,  S 


a,  r,  I 

a,  r,  S 

a,  ^,  S 

r,  ^,  S 


Rcauired. 


Formulfe. 


^^a+(r-l)S 

^(S— ^)"-i_a(S_a)n-i:^0, 
(r— l)Sr«-i 


r'*— 1 


S= 


■    r-l 

rl—a 


Va" 


~y.n y.n-1 


a= 


(r-l)S 


r»— 1  ' 

a=r^— (r— 1)S, 

a(S— a)'^-!— ^(S— ^"-^=0. 


/-=' 


„     S    ,  S— a    _ 
a  ^   a        ' 
S— g 
'S=T» 


log.  ia-^(r—l)S']—log.  a 


log.  r 
log.  I— log.  a 


.4-1, 


log.  (S—a)—lng.  (S -7) 
^Jog.  l~log.  [^r-(r-l)S]    ^ 


GEOMETRICAL  PROGRESSION.  263 

By  observing,  in  any  particular  example,  what  are  given  and  re- 
quired, tlie  proper  formulae  may  be  selected  from  the  above  table. 
Nos.  3,  12,  14,  and  16  may  require  the  solution  of  an  equation 
higher  than  the  second  degree.  Kos.  17,  18,  19,  and  20  are  obtained 
by  solving  an  exponential  equation,  (Art.  382)  but  are  introduced 
here  to  render  the  table  complete.     The  two  formulae 

l=.ar^-^  (1),  and  S  J^£^,  or,  (Art.  298,)  ^    (2\ 

are,  however,  sufficient  for  the  solution  of  all  examples  ia  Geometri- 
cal Progression,  and  may  easily  be  retained  in  the  memory. 

1.  Find  the  8'^  term  of  the  series  5,  10,  20,  etc. 

Aus.  640. 

2.  The  "1^  term  of  the  series  54,  27,  13i,  etc. 

^  Ans.  ||. 

3.  The  6'^  term  of  the  series  3|,  2j,  1^,  etc. 

Ans.  ^. 

4.  The  7^'*  term  of  the  series  —21,  14,  — 9^,  etc. 

Ans.  —  4|f. 

5.  The  7i'*  term  of  the  series  J,  ^,  |,  etc.     Ans.  o^^i* 

6.  Find  the  sum  of  1-f  3-|-9-[-,  etc.,  to  9  terms. 

From  (1),  ^=1X3*^=6561.     From  (2),  S=?^^?^i^=9841,  Ans. 

7.  Of  1+4+16+,  etc,  to  8  terms.      Ans.  21845. 

8.  Of  8  +  20-[-50  +  ,  etc.,  to  7  terms.   Ans.  3249|. 

9.  Of  1  +  3+9+,  etc.,  to  n  terms.     Ans.  ^(3»— 1). 

10.  Of  1— 2+4— 8+,  etc.,  to  n  terms.  Ans.  J(1=f2"). 

11.  Of  X — y-\-- ^+,  etc.,  to  n  terms. 


A„s,4-{i-(-?y{. 


12.  The  first  term  is  4,  the  last  term  12500,  and  the 
^/umber  of  terms  6.  Required  the  ratio  and  the  sum  of 
all  the  terms.  Ans.  Ratio  =^^5 ;  sum  =:15624. 


264  RAY'S  ALGEBRA,  SECOND  BOOK. 

Find  the  sum  of  an  infinite  number  of  terms  of  each  of 
the  following  series : 

13.  Of  5+^  +  1+,  etc Ans.  |. 

14.  Of  9H-6  +  4+,  etc Ans.  27. 

15.  Of|-i  +  ^-    etc Ans.  |. 


16.  Of  a+Z/+-+-2+)  etc An 


a 


a — h 

17.  The  sum  of  an   infinite   geometric  series  is   3,  and 
the  sum  of  its  first  two  terms  is  2| ;  find  the  series. 

Ans.  2+1+1+  •  .  .  or  4-|+|-.  .  .  . 

18.  Find  a  ojeometric  mean  between  4  and  16.    Ans.  8. 


Here,  a=4,  ^=rl6,  and  n=:3;  or,  (Art.  269)  the  mean  =--y^4xl6. 

19.  The  first  term  of  a  geometric  series  is  3,  the  last 
term  96,  and  the  number  of  terms  6 ;  find  the  ratio,  and 
the  intermediate  terms. 

Ans.  r=2.     Int.  terms,  6,  12,  24,  48. 

If  it  be  required  to  insert  m  geometrical  means  between  two 
numbers,  a  and  I,  we  have  n=im-|-2;   hence,  n — l^m-|-l,  and 

r^=::'"+*'y'— .     Or,  we  may  employ  formula  (1). 

20.  Insert  two  geometric  means  between  i^    and  2. 

^Ans.  I,  -^ 

21.  Insert  7  geometric  means  between  2  and  13122. 

Ans.  6,  18,  54,  162,  486,  1458,  4374. 

301*  To  find  the  value  of  Circulating  Decimals;  that 
is,  decimals  in  which  one  or  more  figures  are  continually 
repeated. 

In  such  decimals  the  ratio  is  JL,  y^,  jq^^^.  etc.,  according  as 
one,  two,  or  more  figures  recur.     Thus, 

/  ^1        ^1        ^1  \ 

.253131  ....  =_^»,+  (  _ +jg,  +  j^+,  etc.  ) 


HARMONICAL  PROGRESSION.  265 

The  part  within  the  parenthesis  is  an  infinite  series,  having 
«=T^o^Oo  ^^d  "-=1^    Hence,  (Art.  299,)  S=^^3 ,_. 

Therefore,  .253131  ....  =j%%+^UTi=Ur6=mi- 
This  operation  may  be  performed  more  simply,  as  follows: 

Let S=.25313131     .... 

Multiplying  by  10000,  100008=2531.3131     .     .     . 

Dividing  by  100,      .     .  100S=    25.3131     .     .     . 

Subtracting,    ....  9900S=-2506  .-.  S^fsge. 

1.  Find  the  value  of  .636363 Ans.  j\. 

2.  Find  the  value  of  .54123123.     .     .    Ans.  Jffg^. 

302.  Harmonical  Progression. — Three  or  more  quan- 
tities are  said  to  be  in  Harmonical  Progression,  when  their 
reciprocals  are  in  arithmetical  progression. 

Thus,     1,  |,  ^,  ^,  etc.;  and  1,  2    i,  j^  etc., 
are  in  harmonical  progression,  because  their  reciprocals 

1,  3,  5,  7,  etc.;  and  4,  3|,  3,  2^,  etc., 
are  in  arithmetical  progression. 

303.  Proposition. — If  three  quantities  are  in  harmoni- 
cal progression,  the  first  term  is  to  the  third  as  the  differ- 
ence of  the  first  and  second  is  to  the  difference  of  the  second 
and  third. 

For  if  a,  b,  c,  are  in  harmonical   progression,  — ,  -,  — , 

are  in  arithmetical  progression ;  therefore, 

= = J.     Hence,  multiplying  by  abc, 

ac — bc=r.ab — ac;  or  c(a — b)=a(b — c). 
This  gives  (Art.  268),  a  :  c  :  :  a — b  :  b — c;  therefore, 

A  Harmonical  Progression  is  a  series  of  quantities  in 
harmonical  proportion  (Art.  280)  ;  or  such  that  if  any 
three  consecutive  terms  be  taken,  the  first  is  to  the  third 
as  the  difference  of  the  first  and  second  is  to  the  differ- 
ence of  the  second  and  third. 
2d  Bk.  23* 


266  RAY'S  ALGEBRA,  SECOND  BOOK. 

Hence,  all  problems  with  respect  to  numbers  in  harmon- 
ical  progression,  may  be  solved  by  inverting  them,  and 
considering  the  reciprocals  as  quantities  in  arithmetical 
progression. 

We  give,  however,  below,  two  formulae  of  frequent  use  r 

1.  Given  the  first  two  terms  of  a  harmonical  progres- 
sion, a  and  i,  to  find  the  n^^  term. 

Here,  a,  6,  and  ?,  the  first  two  and  inf^  terms  become  (Art,  302), 

-,     ,  and      in  formula  (1)  (Art.  294).     Also,  C?=t =  — v-' 

Therefore,  _==-4-(n— 1) — ^  =  ^ -^ -\ 

'  I      a^^         '  ab  ab  ' 

Whence,      l=i 


' {n—\)a—{n  -2)6' 

By  means  of  this  formula,  when  any  two  successive  terms  of  a 
harmonical  progression  are  given,  any  other  term  may  be  found. 

2.  Insert  m  harmonic  means  between  a  and  I. 
Here,  since  m^^n — 2,  and  ni-{-l=^n — 1,  we  have,  as  above, 

1  1      ,     .  1XJ  ^     ^  «— ^  «— ^ 

T  =  — \-i^ — l)»j  aiid  <^=7 i — 7  =  7 r-is-  ,1 

I      a^^        '   '  {n—l)al     {m,~\-l)aV 

whence, the  arithmetical  progression  is  found;  and  by  inverting  its 
terms,  the  harmonicals  are  also  found. 

3.  Insert  two  harmonic  means  between  3  and  12. 

Ans.  4  and  6. 

4    Insert  two  harmonic  means  between  2  and  1. 

Ans.  ^  and  §. 

5.  The  first   term   of  a  harmonic  series   is  ^,  and  the 
6'*  is  j'^;  find  the  intermediate  terms. 

6.  a,  J,  c,  are  in  arithmetical  progression,  and  6,  c,  <f, 
are  in  harmonical  progression ;  prove  that  a  :  h  :  :  c  :  d. 


ARITHMETIC  AND  GEOMETRIC  PROGRESSION.       267 


PROBLEMS    IN    ARITHMETICAL    AND    GEOMETRICAL    PRO- 
GRESSION. 

304. — 1.  The  sum  of  5  numbers  in  arithmetical  pro- 
gression is  35,  and  the  sum  of  their  squares  835  ;  find 
the  numbers.  Ans.  1,  4,  7,  10,  13. 

Let  X — 2?/,  X — y^  X,  xyy^  ^+2?/,  be  the  numbers. 

2.  There  are  4  numbers  in  arithmetic  progression,  and 
the  sum  of  the  squares  of  the  extremes  is  68,  and  of  the 
means  52 ;  find  them.  Ans.  2,  4,  6,  8. 

Let  x—Sy,  x—y,  x-{-y,  x-{-Sy,  be  the  numbers. 

Suggestion.— When  the  number  of  terms  in  an  arithmetic 
progression  is  odd,  the  common  difference  should  be  called  2/,  and 
the  middle  term  X\  but  when  the  number  of  terms  is  even,  the  com- 
mon difference  must  be  2y,  and  the  two  middle  terms  x — y  and  x-]-y. 

8.  The  sum  of  3  numbers  in  arithmetical  progression 
is  80,  and  the  sum  of  their  squares  308 ;  find  them. 

Ans.  8,  10,  12. 

4.  There  are  4  numbers  in  arithmetical  progression, 
their  sum  is  26,  and  their  product  880  ;  find  them. 

Ans.  2,  5,  8,  11. 

5.  There  are  3  numbers  in  geometrical  progression,  whose 
sum  is  31;  and  the  sum  of  the  1st  and  2d  :  sum  of  1st 
and  3d  :  :  3  :  13  ;  find  them.  Ans.  1,  5,  25. 

Eet  x=  1st  term  and  y=^  ratio;  then,  xy  and  xy^=  2d  and  3d 
terms. 

6.  The  sum  of  the  squares  of  3  numbers  in  arithmetical 
progression  is  83 ;  and  the  square  of  the  mean  is  greater 
by  4  than  the  product  of  the  extremes ;  find  them. 

Ans.  3,  5,  7. 

7.  Find  4  numbers  in  arithmetical  progression,  such  that 
the  product  of  the  extremes  =27  ;  of  the  means  =r35. 

Ans.   3,  5,  7    9. 


268  RAYS  ALGEBRA,  SECOND  BOOK. 

8.  There  are  3  numbers  in  arithmetical  progression, 
whose  sum  is  18;  but  if  you  multiply  the  first  term 
by  2,  the  second  by  3,  and  the  third  by  6,  the  products 
will  be  in  geometrical  progression ;  find  them. 

Ans.  3,  6,  9. 

9.  The  sum  of  the  fourth  powers  of  three  successive 
natural  numbers  is  962  ;  find  them.  Ans.  3,  4,  5. 

10.  The  product  of  four  successive  natural  numbers  is 
840 ;  find  them.  Ans.  4,  5,  6,  7. 

11.  The  product  of  four  numbers  in  arithmetical  pro- 
gression is  280,  and  the  sum  of  their  squares  166;  find 
them.  Ans.  1,  4,  7,  10. 

12.  The  sum  of  9  numbers  in  arithmetical  progression 
is  45,  and  the  sum  of  their  squares  285;  find  them. 

Ans.  1,  2,  3,  etc.,  to  9. 

13.  The  sum  of  Y  numbers  in  arithmetical  progression 
is  35,  and  the  sum  of  their  cubes  1295;  find  them. 

Ans.  2,  3,  etc.,  to  8. 

14.  Prove  that  when  the  arithmetical  mean  of  two  num- 
bers is  to  the  geometric  mean  :  :  5  :  4 ;  that  one  of  them 
is  4  times  the  other. 

15.  The  sum  of  3  numbers  in  geometrical  progression 
is  7 ;  and  the  sum  of  their  reciprocals  is  J ;  find  them. 

Ans.  1,  2,  4. 
Suggestion. — In  solving  difficult  problems  in  geometrical  pro- 
gression, it  is  sometimes  preferable  to  express  them  by  other  forms. 

xi 

Thus,  for  3  numbers,  use  X,  y/xy^  y^  or,  a:^,  xy^  y^;  for  four,.  — , 

X,  y,  ^-;   for  five,  -,  x^,  xy,  y\  |-;  for  six,  ^,  _,  x,  y,  |-,  ^. 

In  all  these  cases  the  product  of  the  first  and  third  of  any  three, 
taken  consecutively,  is  equal  to  the  square  of  the  second.  To  find 
the  ratio  in  each  case,  divide  any  expression  by  the  preceding. 

16.  There  are  4  numbers  in  geometrical  progression, 
the  sum  of  the  first  and  third  is  10,  and  the  sum  of  the 
second  and  fourth  is  30;  find  them.      Ans.  1,  3,  9,  27. 


PERMUTATIONS  AND  COMBINATIONS.  269 

17.  There  are  4  numbers  in  geometrical  progression,  the 
sum  of  the  extremes  is  35,  the  sum  of  the  means  is  30; 
find  them.  Ans.  8,  12,  18,  27. 

18.  There  are  4  numbers  in  arithmetical  progression, 
which  being  increased  by  2,  4,  8,  and  15  respectively, 
the  sums  are  in  geometrical  progression ;  find  them. 

Ans.  6,  8,  10,  12. 

19.  There  are  3  numbers  in  geometrical  progression, 
whose  continued  product  is  64,  and  the  sum  of  their 
cubes  584;  find  them.  Ans.  2,  4,  8. 


IX.  PERMUTATIONS,  COMBIJ^^ATIOIS^S, 
AND  BINOMIAL  THEOREM. 

305.  The  Permutations  of  quantities  are  the  different 
orders  in  which  they  can  be  arranged. 

Quantities  may  be  arranged  in  sets  of  one  and  one,  two 
and  two,  three  and  three,  and  so  on. 

Thus,  if  we  have  three  quantities,  a,  h,  c,  we  may  arrange 
them  in  sets  of  one,  of  two,  or  of  three,  thus  : 

Of  one,  a,  h,  c. 

Of  two,  ah,  ac;       ha,  he,       ca,  ch. 

Of  three,  ahc,  ach ;    hac,  hca ;  cah,  cha. 

306.  To  find  the  number  of  permutations  that  can  be. 
formed  out  of  n  letters,  taken  singly,  taken  two  together, 
three  together.  .  .  .  and  r  together. 

Let  a,  b,  c,  d,  .  .  .  .  k,  be  the  n  letters ;  and  let  P^  denote  the 
whole  number  of  permutations  where  the  letters  are  taken  singly; 
Pg  the  whole  number,  taken  2  together  ....  and  P,.  the  number 
taken  r  together. 


270  RAY'S  ALGEBRA,  SECOND  BOOK. 

The  number  of  permutations  of  n  letters  taken  singly,  is  evi- 
dently equal  to  the  number  of  letters;  that  is, 

The  number  of  permutations  of  n  letters,  taken  two  together,  is 
n{n—\).     For  since  there  are  n  quantities, 

a,  b,  c,  d,    .    .    .    .    Je, 

if  we  remove  a,  there  will  remain  [n—l)  quantities.     Writing  a 
before  each  of  these  [n — 1)  quantities,  we  shall  have 
ab,  ac,  ad    ...    .    ak. 

That  is,  [n — 1)  permutations  in  which  a  stands  first. 

In  the  same  manner,  there  are  {n—\)  permutations  in  which  b 
stands  first,  and  so  of  each  of  the  remaining  letters  C,  d,  .  .  .  k. 
Or,  for  n  letters,  there  are  n[n — 1)  permutations  taken  two  together. 
That  is,  V2=n{n—1).     Hence, 

The  number  of  permutations  of  n  letters  taken  two  together, 
is  equal  to  the  number  of  letters^  multiplied  by  the  number 
less  one. 

For  example,  if  w=4,  the  number  of  permutations  of  Cf,  b,  c,  d, 
taken  two  together,  is  4x3=12.  Thus,  ab,  ac,  ad,  \\  ba,  be,  bd, 
II  ea,  cb,  cd,  \\  da,  db,  do. 

The  number  of  permutations  of  n  letters,  taken  three  together,  is 
n{n — l)(n— 2).     For  if  we  take  (n—l)  letters, 

b,  c,  d,      .      ...      k,  the  number  of  permu- 
tations taken  two  together,  by  the  last  paragraph,  is 
(n— l)(n-2). 

Let  a  be  placed  before  each  of  these  permutations ;  then,  there 
are  (n — l)(n— 2)  permutations  of  n  letters,  taken  three  together,  in 
which  a  stands  first,  and  {n — l)(n— 2)  permutations,  in  which  b 
stands  first ;  and  so  for  each  of  the  n  letters. 

Hence,  the  whole  number  of  permutations  of  n  letters,  taken  three 
together,  is  n(n— l)(n— 2); 

That  is,  P3=n(n— 1 ) (n— 2).     Hence, 

7%e  number  of  permutations  of  n  letters  taken  three  to- 
gether^ is  equal  to  the  number  of  letters,  multiplied  by  ike 
number  less  one,  multiplied  by  the  number  less  two. 


PERMUTATIONS  AND  COMBINATIONS.  271 

If  n=4,  the  number  of  permutations  of  a,  6,  c,  d,  taken  three 
together,  is  4x3x2  ==24.     Thus, 

abc,  abd,  acb,  acd,  adb,  adc,  bae,  bad,  bca,  bed,  bda,  bde, 

cab,  cad,  cba,  cbd,  cda,  cdb,  dab,  dac, 

dba,  dbe,  dca,  deb. 

Following  the  same  method,  we  prove  that  the  number  of  permu- 
tations of  n  letters  taken  four  together,  is 

P4=n(n— l)(w— 2)(n— 3). 

In  each  of  the  preceding  results,  the  negative  number  in  the  last 
factor  is  less  by  unity,  than  the  number  of  letters  in  each  permu- 
tation. 

Hence,  for  n  things  taken  r  together, 


P^=n(n— l)(n— 2) (n— r— 1) 

306^.  Corollary. — If  all  the  letters  be  taken  together, 
then  r  becomes  equal  to  n,  and  the  last  factor  becomes  1  ; 

That  is,  P„=«(«— !)(«— 2) 1. 

Or,  inverting  the  order  of  the  factors, 

P„=:lx2x3 (71—1)71.     Hence, 

Tlie  number  of  permutations  of  n  letters  taken  n  together^ 
is  equal  to  the  product  of  the  natural  numbers  from  1  up 
to  n. 

Ex. — The  permutations  of  three  letters,  a,  6,  c,  taken 
three  together,  is  1x2x3=6. 

307*  If  the  same  letter  occur  p  times,  the  !i::mber  of 
permutations  in  n  letters,  taken  all  together,  is 

1X2X3   ....   (n-l)n 
1X2X3. . .p 

Suppose  these  p  letters  to  be  all  different.     Then,  for  any  psrilo 
ular  position  of  the  other  letters,  these  p  quantities,  taken  p  to- 
gether, will  form   (1x2X3    .    .    .    .   p)  permutations  from  their 
interchange  with  each  other;  and  when  these  letters  are  alike,  these 
permutations  are  all  reduced  to  one.     And 


272  RAY'S  ALGEBRA,  SECOND  BOOK. 

As  this  is  true  for  every  position  of  the  other  letters,  there  will 
be  altogether  (1X2X3  •  -  -  P)  times  fewer  permutations  when  they 
are  alike  than  when  they  are  all  different. 

Thus,  in  the  letters  A,  I,  D,  there  are  1X2X^=6  permutations 
taken  all  together,  but  if  I  becomes  D,  then  three  of  these  permuta- 
tions become  identical  with  the  remaining  three,  and  the  whole 
number  for  ADD  taken  all  together,  is 

"1X2-=^- 

307^  Corollary. — In  like  manner,  if  the  same  letter 
occur  p  times,  another  letter  q  times,  a  third  letter  r  times, 
and  so  on,  the  number  of  permutations  taken  all  to- 
gether, is 

1X2X3 (n-l)n 

(1X2  .  .  i>)(lx2  .  .  2)(1X2  .  .  r)x,  etc.  * 

308.  The  Combinations  of  quantities  are  the  different 
collections  that  can  be  formed  out  of  them,  without  refer- 
ence to  the  order  in  which  they  are  placed. 

Thus,  a6,  ac,  he,  are  the  combinations  of  the  letters  a, 
5,  c,  taken  two  together;  ah  and  ha,  ac  and  ca,  he  and  c6, 
though  diflFerent  permutations,  forming  the  same  combina- 
tion. 

Proposition. —  To  find  the  number  of  comhinatiom  that 
can  he  formed  out  of  n  letters,  taken  singly,  taken  two  to- 
gether, three  together, and  r  together. 

Let  C„  Cj,  .  .  .  Or  denote  the  number  of  combinations 
of  n  things  taken  singly,  taken  two  together,  ....  and 
taken  r  together. 

The  number  of  combinations  of  n  letters  taken  singly,  is  evi- 
dently n;  that  is, 

C,=n. 

The  number  of  permutations  of  n  letters  taken  two  together,  is 
rt{n — 1);  but  each  combination,  as  a6,  admits  of  (1X2)  permuta- 


C2=- 


PERMUTATIONS  AND  COMBINATIONS.  273 

tions,  a6,  ha]  therefore,  there  are  (IX^)  times  as  many  permuta- 
tions as  combinations.     Hence, 

n(n— 1) 
"  Tx2"- 

Again,  in  n  letters  taken  three  together,  the  number  of  permuta- 
tions is  n{n — l)(n— 2);  but  each  combination  of  three  letters,  as 
abc,  admits  of  1x2x3  permutations;  therefore, 

_n{n-l){n-2) 
^3-      1^2X3      • 

So,  for  n  letters,  each  of  which  contains  r  combinations, 
n(?8~l)(n-2)  .  .  .  [n-(r-l)] 
'~        1X2X3 r 

300.  Intimately  connected  with  the  subject  of  the  pre- 
ceding articles,  is  that  of  the  Doctrine  of  Chances,  or 
the  Calculus  op  Probabilities.  This,  however,  being 
too  abstruse  for  an  elementary  treatise,  is  omitted  in  this 
work. 

1.  How  many  permutations  of  2  letters  each  can  be 
formed  out  of  the  letters  a,  6,  c,  t/,  e?  How  many  of  3? 
Of  4?  Ans.  (1)  20.  (2)  60.  (3)  120. 

2.  How  many  combinations  of  2  letters  each  can  be 
formed  out  of  the  letters  a,  6,  c,  d,  e?  How  many  of  3? 
Of  4?    Of  5?  Ans.  (1)  10.  (2)  10.  (3)  5.  (4)  1. 

3.  In  how  many  ways,  taken  all  together,  may  the  letters 
in  the  word  NOT  be  written  ?     In  the  word  HOME  ? 

Ans  6,  and  24. 

4.  How  often  can  6  persons  change  their  places  at  din- 
ner, so  as  not  to  sit  twice  in  the  same  order?    Ans.  720. 

5.  In  how  many  different  ways,  taken  all  together,  can 
the  7  prismatic  colors  be  arranged?  Ans.  5040. 

6.  In  how  many  different  ways  can  6  letters  be  arranged 
when  taken  singly,  2  by  2,  3  by  3,  and  so  on,  till  they  are 
all  taken?  Ans.  1956. 

Suggestion . — Take  the  sum  of  the  different  permutations. 


274  RAY'S  ALGEBRA,  SECOND  BOOK. 

7.  How  many   different  products  can   be  formed  with 
any  two  of  the  figures  3,  4,  5,  G  ?  Ans.  6. 

8.  The  number  of  permutations  of  n  things  taken  4  to- 
gether =  6  times  the  number  taken  3  together ;  find  n. 

Ads.  n=0. 

9.  How  many  different  sums  of  money  can  be  formed 
with  a  cent,  a  three  cent  piece,  a  half  dime,  and  a  dime  ? 

Ans.  15. 

Suggestion. — Take  the  sum  of  the  different  combinations  of 
4  things  taken  singly,  2  together,  3  together,  and  4  together. 

10.  With  the  addition  of  a  twenty-five  cent  piece,  and  a 
half  dollar,  to  the  coins  in  the  last  example,  how  many 
different  sums  of  money  may  be  formed?  Ans.  63. 

11.  At  an  election,  where  every  voter  may  vote  for  any 
number  of  candidates  not  greater  than  the  number  to  be 
elected,  there  are  4  candidates  and  only  3  persons  to  be 
chosen  ;  in  how  many  ways  may  a  man  vote?     Ans.  14. 

12.  On  how  many  nights  may  a  different  guard  of  4  men 
be  posted  out  of  16?  and  on  how  many  of  these  will  anj 
particular  man  be  on  guard?  Ans.  1820,  and  455. 

13.  How  many  changes  may  be  rung  with  5  bells  out 
of  8,  and  how  many  with  the  whole  peal? 

Ans.  6720,  and  40320. 

14.  Out  of  l7  consonants  and  5  vowels,  how  many  wordil 
can  be  formed,  having  two  consonants  and  one  vowel  in 
each?  Ans.  4080 


BINOMIAL    THEOREM, 

WHEN    THE     EXPONENT    IS    A    POSITIVE    INTEGER. 

310.  We  have  already  explained  (Art.  172)  the  method 
of  finding  any  power  of  a  binomial,  by  repeated  multiplica- 
tion, and  by  Newton's  Theorem,  as  proved  experimentally. 


BINOMIAL  THEOREM. 


275 


We  shall  now  proceed  from  the  theory  of  Combinations 
(Art.  308),  to  demonstrate  the  Binomial  Theorem  in  its 
most  general  form. 

The  Binomial  Theorem  teaches  the  method  of  develop- 
ing into  a  series  any  binomial  whose  index  is  either  in- 
tegral or  fractional,  positive  or  negative ;  as, 

(a+x)",   (a+a.)-^  {a-\-xy\  (a-^x)-\ 
where  a  or  x  may  be  either  plus  or  minus. 

The  following  investigation  applies  only  to  the  case 
where  the  exponent  is  positive  and  integral;  the  other  cases 
will  be  considered  hereafter.     (See  Art.  319.) 

By  actual  muhiplication,  it  appears  that 

{x-{-a){x-\-b)—x^-\^a  i  x^ah. 
+  61 

In  like  manner,  {x-\-a){x-^b){x-\-G) 


+6 


x'^-\-ab  x-\-abc. 
-f  «c 
-f6c 


Also,  {x^a){x-{-b)(x^c){x-\-d) 


=zx*-\-a 

x^-\-ab 

x'^+abc 

4-6 

-^ac 

-\-abd 

-fc 

-fad 

-\-acd 

J,d 

-f-6c 

-{-bed 

-\-bd 

-fccZj 

x-\-abcd. 


An  examination  of  either  of  these  products,  shows  that  it  is  com- 
posed of  a  series  of  descending  powers  of  X,  and  of  certain  coeffi- 
cients, formed  according  to  the  following  law: 

1st.  The  exponent  of  the  highest  power  of  x  is  found  in 
the  first  term,  and  is  the  same  as  the  mtmher  of  binomial 
faetors,  and  the  other  exponents  of  x  decrease  by  1  in  each 
succeeding  term, 

2d.  The  coefficient  of  the  first  term  is  1 ;  of  the  second, 
the  sum  of  the  quantities  a,  6,  c,  etc. ;  of  the  third,  the 


276  RAY'S  ALGEBRA,  SECOND  BOOK. 

sum  of  the  products  of  every  two  of  the  quantities  a,  &,  c, 
etc. ;  of  the  fourth,  the  sum  of  the  products  of  every  three, 
and  so  on ;  and  of  the  last,  the  product  of  all  the  n  quan- 
tities a,  b,  c,  etc. 

Suppose,  then,  this  law  to  hold  for  the  product  of  n  binomial  fac- 
tors a;+ a,  x-\-b,  ^+c, x-[-k-  so  that  {x-{-a){x-^b){x-^c) 

(a;-f  ^)=^-a;^^4-Aa;''-i4-Ba;"-2+Ca;"-3+ -f  K, 

Where    .    .    ,         A=a-f-6+c+  ....  -f  A;. 

B=ab-\^ac-\-ad-{- 

C=a6c-f  a6cZ-f- 

Etc.  =  etc 

Y.=^abcd Jc. 

If  we  multiply  both  sides  of  this  equation  by  a  new  factor  x-\-lj 

we  have 

{x-^a){x-\-b){x-j-c) {xi-'k){x-\-l) 

j^l   I    _^a;  I       -I-B;  1  ....  +KZ. 

Here,     .     .     A+^  =a+6+c+  »  .  .  .  +7^+?; 

B-f  A^  =ab+ac-{-ad .  .  .  -^al-\-bl  .  .  .  -j-Jcl. 

Etc.  =  etc 

Kl  =abcd JcL 

It  is  evident  that  the  same  law,  as  above  stated,  still  holds. 

Hence,  if  the  law  holds  when  n  binomial  factors  are  multiplied 
together,  it  will  hold  when  n-|-l  factors  are  taken;  but  it  has  been 
shown,  by  actual  multiplication,  to  hold  up  to  4  factors;  therefore, 
it  is  true  for  4+1,  that  is,  5;  and  if  for  6,  then  for  5-[-l,  that  is,  6; 
and  so  on  generally,  for  any  number  whatever. 

Now  let     .     .     .     b,  c,  d,  etc.,  each  —a; 

Then,    A=a-f  a+a-f-a+j  etc.,  to  n  terms  =na. 

B=a^-\-a'^-{-,  etc.,  =a2  taken  as  many  times  as  ^ 
is  equal  to  the  No.  of  combinations  of  n  things  taken  j-  =  . —  . 

two  together,  which  is  (Art.  308),  ^ 

C=a3-|-a3_[_^    etc.,    =a^   taken   as   many 
times  as  is  equal  to  the  No.  of  combinati 
of  the  things  taken  three  together,  which 
(Art.  308), 

Etc.  =  et6. 
K=zaaa  ....  to  n  factors  ^a" 


aany  n 

Lions  (  _n(n—l)(n—2)a^ 

;hisj  1-2-3       ~* 


BINOMIAL  THEOREM.  277 

Also,  (a:-fa)(a;+6)(a;+c) {x^l)={x-\-aY. 

+ +«". 

By  changing  a:  to  <x  and  a  to  x,  we  have 

n{n-l){n-2)  

Let  a=:l ;  then,  since  every  power  of  1  is  1, 

nin — 1)  „    n{n—\)(n — 2)  „  , 
(l+a;)"^Hna;+-L-^a;24-   ^    ^J^^^      ^x^J^ J^x-. 

Corollary  1. — The  sum  of  the  exponents  of  a  and  x  in 
each  term  =^n. 

Corollary  2. — If  either  term  of  the  binomial  is  negative, 
every  odd  power  of  that  term  will  be  negative  (Art.  193); 
therefore,  the  terms  which  contain  the  odd  powers  will  be 
negative. 

n(n—l)  „    nln—\)(n — 2)  „ 
...  (1— a;)»=l— na:+   ^         'x^ ^    1.2-3 '^'  ®*^' 

Corollary  3. — Since  the  last  factors,  in  the  fraction 
which  forms  the  coefl&cient,  are  for  the  2d  term  y,  for  the 

??— 1  71—2 

3d  term  — j^— ,  for  the  4th  term  — ^-j  etc.;  therefore,  for 

^_(^_2) 
the  r^^  term  they  will  be  ^-^ 

Also,  for  the  exponents  of  a  and  a;,  we  have  in  the  2d 
term  a"-\'c,  in  the  3d  term  a"-V,  in  the  4th  term  a^-^r^; 
therefore,  in  the  r^^  term,  we  shall  have  a"-^'-^V-^ 

Hence,  the  general  term  of  the  series  is 

n(n-l)(n-2) iri-~r-^2)  _^ 

— 1-2-3 (^=iy~         * 


278  RAY'S  ALGEBRA,  SECOND  BOOK. 

This  is  called  the  general  term,  because  by  making  r=2, 
8,  4,  etc.,  all  the  others  can  be  deduced  from  it. 

Example. — Required  the  W^  term  of  (a — xy. 
Here,  r=:5,  and  71=^7;  therefore,  the  term  required 
4  .  5  .  6  .7 


1  •2-3-4 


.{af[—xY=.2>^a^xK 


Corollary  4. — If  n  be  a  positive  integer,  and  r=n-\-2  ; 
then,  {n — r-\-2)  becomes  0,  and  the  (n-|-2)  term  vanishes; 
therefore,  the  series  consists  of  (n-\-V)  terms  altogether; 
that  is,  • 

The  number  of  terms  is  one  greater  than  the  exporient  of 
the  power  to  ivhich  the  binomial  is  to  be  raised. 

Corollary  5. — When  the  index  of  the  binomial  is  a  posi- 
tive integer,  the  coefficients  of  the  terms  taken  in  an  in- 
verse order  from  the  end  of  the  series,  are  equal  to  the 
coefficients  of  the  corresponding  terms  taken  in  a  direct 
order  from  the  beginning. 

If  we  compare  the  expansion  of  (a-^x)^,  and  (a:-f  a)"",  we  have 

{xJrar=x--^nx--^a-\-'^-^^=^ 

Since  the  binomials  are  the  same,  the  series  resulting  trom  their 
expansion  must  be  the  same,  except  that  the  order  of  the  terms  will 
be  inverted.  It  is  clearly  seen  that  the  coefficients  of  the  corre- 
sponding terms  are  equal. 

Hence,  in  expanding  such  a  binomial,  the  latter  half  of  the  ex- 
pansion may  be  taken  from  the  first  half. 

Example. — Expand  (a — Z>)^ 

Here  the  number  of  terms  (n-fl)  is  6;  therefore,  it  is  only  neces- 
sary to  find  the  coefficients  of  the  first  three,  thus: 

(a— 6)5=a5_5«45^^(^362_i0a2^.|.5«64_5.'>. 


BINOMIAL  THEOREM.  279 

Corollary  6. — The  sum  of  the  coefl5cients,  where  both 
terms  are  positive,  is  always  equal  to  2'*. 

For  if  we  makea;=a=l;  then,    .     .         ra;+a)»=r(l+l)»»=2". 

311.  From  an  inspection  of  the  general  expansion  of 

(a-f  x)",  it  is  evident  that 

If  the  coefficient  of  any  term  he  multiplied  hy  the  expo- 
nent of  the  first  letter  of  the  binomial  in  that  term,  and  the 
product  he  divided  hy  the  number  of  the  term,  the  quotient 
will  be  the  coefficient  of  the  next  term. 

For  examples,  see  Newton  s  Theorem,  Art.  172. 

31!3.  To  expand  a  binomial  aifected  with  coefficients 
or  exponents,  as  (2a'^  —  86^)*,  see  Newton  s  Theorem, 
Art.  172. 

313.  By  means  of  the  Binomial  Theorem,  we  can  raise 
any  polynomial  to  any  power.  Thus,  let  it  be  required  to 
raise  a—h-\-c  to  the  third  power. 

Let  a — b=m,  etc.,  as  already  explained,  Art.  172. 

1.  Expand  (a+5)«,  (a—hy,  and  (5— 4a-)*. 

(1)  Ans.  a'-\-Sa'b-\-2Sa'b'Jr^Qc^'f>'-\-l0a'b'-^b6a'h^ 

Jr2Sa'h^~\-SaP-\-h\ 

(2)  Ans.  a''~7a'b-^21a'b'—S^a'L'-\-Sba^b*~21a'h' 

j^^ah^—h\ 

(3)  Ans.  625— 2000a- -h 2400x^—1 280a:3-f  256a;*. 

2.  Kequired  the  coefficient  of  x^  in  the  expansion  of 
(^-j_y)io.  Ans.  210. 

3.  Find  the  5'^^  term  of  the  expansion  of  (c^ — d^y^. 

Ans.  495c^«<Z«. 

Suggestion  .—(See  Cor.  3,  Art.  310.)  Instead  of  a,  x,  n,  and  r, 
substitute  c2,  —d-,  12,  and  5. 

4.  Find  the  7"^  term  of  (a»+3a?.)».     Ans.  61236a^55«. 


280  RAY'S  ALGEBRA,  SECOND  BOOK. 

5.  Find  the  middle  term  of  (a^'-^-xy.    A.  924a«'"a^'*. 

6.  Find  the  8^''  term  of  (1-f  a^)".  Ans.  330x\ 

7.  Expand  (Sac—2bdy.         Ans.  243aV— 810aV6cZ 

-\-10SQa'c'h'd'—l20a'c'b^d^-\-240ac¥d*~S2h'd'. 

8.  Expand  (a-^2b—cy.  Ans.  a34-6a'^64-12at2 

+  86'— Sa^c— 12a&c— 12Z>2c+3ac2+6fec='— c'. 

9.  Prove  that  the  sum  of  the  coefficients  of  the  odd  terms 
of  (a-f  ^)"»  is  equal  to  the  sum  of  the  coefficients  of  the 
even  terms. 


X.  INDETERMINATE  COEFFICIENTS:  BINOMIAL 
THEOREM,  GENERAL  DEMONSTRATION : 
SUMMATION  AND  INTERPOLA- 
TION OF  SERIES. 

314.  Indeterminate  Coefficients.—  The  method  of  de- 
veloping algebraic  expressions,  by  assuming  a  series  with 
unknown  coefficients,  and  finding  the  values  of  the  assumed 
coefficients,  is  termed  the  method  of  Indeterminate  Coejfi- 
cients.     It  depends  on  the  following 

THEOREM. 

If  A+Br^-fCa^^+Dor'-f,  etc  ,  =rA'+B'ic-f  CV4  DV+, 
etc.,  for  every  possible  value  of  x  (A,  B,  A',  B',  etc.,  not 
containing  x,  and  x  being  a  variable  quantity)  we  shall 
have  A=A',  B=B',  C=C',  etc.;  that  is, 

The  coefficients  of  the  terms  involving  the  same  powers  of  x 
in  the  two  series,  are  respectively/  equal. 

For,  by  transposing  all  the  terms  into  the  first  member,  we  have 
A—A'-\-{B—B')x-\-{C—C')x^-\-iJ)—T)')x^-\-,  etc.,  =.0. 

If  A — A^  is  not  equal  to  0,  let  it  be  equal  to  some  quantity  p; 
then,  we  have  {B—B')x^{C—C')x^^{D—Ty)x^-j-^  etc.,  =—p. 


INDETERMINATE  COEFFICIENTS.  281 

Now,  since  A  and  Af  are  constant  quantities,  their  difference,  p^ 
must  be  constant;  but  — p=(B— B^)a:-|-(C— C^)a:2+,  ^xe.,  a  quan- 
tity which  may  evidently  have  various  values,  since  it  depends 
upon  X]  therefore,  the  same  quantity  [p)  is  both  fixed  and  variable, 
which  is  impossible. 

Hence,  there  is  no  possible  quantiiy  (p)  which  can  express  the 
difference  A— A^;  or,  in  other  words, 

A— A^=0    .-.     Arr:A^ 

Hence,  {B—B')x-\-{C—C')x^^{I)—jy)x^-j-,  etc.,  =0. 

By  dividing  each  side  by  x,  we  have 

B-B'+(C— C0a:+(D-iy)a:2-f,  etc.,  =0. 

Reasoning  as  before,  we  may  show  that  B^iB';  and  so  on,  for  the 
remaining  coefficients  of  the  like  powers  of  X. 

Corollary. — If  we  have  an  equation  of  the  form 

A^Bx-\-Cx'-^Dx'-\-Ex'-\-,  etc.,  ==0, 

which  is  true  for  avi/  value  whatever  of  x;  then,  Arrr^O, 
B=0,  C±=0,  etc.;  that  is,  each  coefficient  is  separately  equal 
to  zero. 

For  the  right  hand  member  may  evidently  be  put  under  the  form 
0-f0a:-(-0a:2-^0a:3-[-,  etc.;  then,  comparing  the  coefficients  of  the  like 
powers  of  x,  we  have  A=0,  Bz=lO,  C=0,  etc. 

31S.  Let  it  be  required  to  develope  — —7—  into  a  series 

a-\-ox 

without  a  resort  to  division. 

The  series  will  consist  of  the  powers  of  X  multiplied  by  certain 
undetermined  coefficients,  depending  on  either  a  or  6,  or  both  of 
them,  and  x  will  not  enter  into  the  first  term;    therefore,  assume 

^^=A+Ba:+Ca:2+Da:3+,  etc. 

Multiply  both  sides  by  the  denominator  a-\-hx,  and  arrange  the 
terms  according  to  the  powers  of  x\  we  thus  obtain 

a=ka-\-Ba  I  x-\-Qa  I  x'^-\-Da  I  x^-\-^  etc. 
+A6|    +B6|      +C6| 
2d  Bk.        24 


282  RAY'S  ALGEBRA,  SECOND  BOOK. 

But  by  the  preceding  theorem  and  corollary, 

a=Aa  ;  hence,  A^l ; 

Ba+A6=0:        "        B= ; 

a' 

Ca+B6=0;       «       C=:+^,; 
53 

Substituting  these  values  in  the  assumed  series,  we  find 

a  b       b^        b^        b^ 

1 X-}-  —x^ :.x^-] — -x^,  etc.,  the  sa'me  as  would 


a^bx  a      a'^        a^        a* 

be  obtained  by  actual  division. 

316.  A  series  with  indeterminate  coefficients  is  gener- 
ally assumed  to  proceed  according  to  the  ascending  in- 
tegral and  positive  powers  of  x,  beginning  with  x° ;  but  in 
many  series  this  is  not  the  case.  The  error  in  the  assump- 
tion will  then  be  shown,  either  by  an  impossible  result, 
or  by  finding  the  coefficients  of  the  terms  which  do  not 
exist  in  the  actual  series,  equal  to  zero. 

Thus,  if  it  be  required  to  develope  ^ 5'   ^^^^  ^^  assume  the 

series  to  be  A+Bic-f  CiC^-j-DrcS-f  EiC*-]-,  etc.,  we  have,  after  clearing 
of  fractions, 

l=3Aa;-h(3B— A)a:2-f  (3C— B)a:34.,  etc.; 

from  which,  by  equating  the  coefficients  of  the  same  powers  of  x, 

1=0,  3A=0,  etc. 

The  first  equation,  1==0,  being  absurd,  we  infer  that  the  expres- 
sion can  not  be  developed  under  the  assumed  form.     But, 

clearing    of  fractions,  and   equating  the  coefficients  of  the   like 
powers  of  X,  we  find  A=|,  B=^,  C^^i^,  D=g'p  etc.     Hence, 


Sx 


1  1/1      X      x\  x^\  \     x-\x\    x       x\       ^ 


INDETERMINATE  COEFFICIENTS.  283 

Or,  since  the  division  of  1  by  the  first  term  of  the  denominator 

gives  — ,  or  Sx~^,  we  ought  to  have  assumed 
oX 

Ax-^-\-B-\-Cx-\-'Dx-^,  etc. 


Sx — x'^ 


1 x2 

Again,  if  we  assume  ^ ^=A-fBa;+Ca:2-f  Da:3-f ,  etc.; 

we  shall  find  the   true   series  to  be  l—2x--\3x*—5x^-\-,  etc.,  the 
coefficients  B,  D,  F,  etc.,  becoming  zero. 


317*  Evolution  by  indeterminate  coefficients. 
Example. — Extract  the  square  root  of  a^-\-x^. 

Assume  y {a'^^x'^)=k^J^x-\-Qx--\-J)x^^^x^^^  etc. 
Squaring  both  sides,  we  have, 

a2_|_a;2^A2-f  2ABa;+2AC  i  a;2-]-2AD  1  a:34-2AE   x^-\-,  etc. 
4-B2  I      -f  2BC  I      +2BD 

+  02 

.-.  A2=:a2,  2AB=.0,  2AC+B2==1,  2AD+2BC=0,  etc. 
And,   A=a,   B=0,    C^^-'  ^=0,  Er=— — ^  — ,  etc. 
Therefore,  v/(«2-t-a;2)=a-{-|-_^34-,  etc 

SIS.  Decomposition  of  Rational  Fractions. — Frac- 
tions whose  denominators  can  be  separated  into  certain 
factors,  may  often  be  decomposed  into  other  fractions 
whose  denominators  shall  consist  of  one  or  more  of  these 
factors.     To  illustrate  by  an  example. 

^x 14 

Decompose     .^  '  ^ — — ^  into  two   other  fractions   whose 

denominators  shall  be  the  factors  of  x^ — 6;r-f8. 

Since  x'^—^x-\-^={x^2){x-^\  (Art.  234),  assume 
5a;_14    _    A  B 


284  RAY'S  ALGEBRA,  SECOND  BOOK. 

Beducing  the  fractions  to  a  common  denominator, 
We  have  5:r-14    _A(a:-4)+B(:r.-2). 

Or,        ox—U=A{x—A)-{-B{x—2)={A-\-B)x—4A—2B. 

Now,  since  this  equation  is  true  for  any  value  whatever  of  X,  we 
may  equate  the  coefficients  (Art.  314);  this  gives 

A-|-B^5;  _4A— 2B=— 14;  whence,  A=2,  and  B^^. 
And     .    .    .    ^2_Qx-{-8~~x~2'^x^' 

By  the  method  of  Indeterminate  Coefficients,  show  that 

2.  l+2x  ^^i^3^^4^2_^7^8^11^4_^18^5^^  etc. 

J. X X 

3.  n_l"^)3"^l'+^'^H-3V-{-4V-f5V+,  etc. 
cc  x^  Sa^  S'bx* 


4.  i/l— ^=1-2~2~^""2^1^~2~T^8~  ^*^- 

5.  ^/(l+^+^^):=l+|+|!_|^^+,  etc. 
«    1+^      1  ,     2 


X X"  X 


a;-'— 7x+12~~x— 4     x— 3* 

8.  ,-,_,^;_,,=  ^^--i   +   1 


(x^— 1X^5—2)  ~  3(0;— 2)      2(a;— 1)  "^  6(a;H-l)' 


BINOMIAL  THEOREM.  285 

BINOMIAL    THEOREM, 

WHEN    THE    EXPONENT    IS    FRACTIONAL    OR    NEGATIVE. 

SIO.  We  shall  now  proceed  to  prove  the  truth  of  the 
Binomial  Theorem  generally ;  that  is,  to  show  that 

^\'hethe^  n  be  integral  or  fractional,  positive  or  negative. 
Since a-\-b=a{  \-\ —  ); 

Therefore,  (a-f  6)"=a''  /  l-j—  )  ''^a'\\-\-x)\  if  x=~. 

Hence,  if  we  can  find  the  law  of  the  expansion  of  {\-\-x)^  we  may 

obtain  that  of  (a-J-fe)",  by  writing  -  for  a:,  and  multiplying  by  a". 

a 

AVe  shall  therefore  prove  that,  in  all  cases, 

/i  I     X,.     1  I        .  ^(^—1)   9  ,  n(n— l)(n— 2)  ,  , 
(l-fa:)"=l-i-na:-|-A___^a:2-f -A-^-^^-^ — 'x^J^,  etc. 

The  proof  may  be  divided  into  two  parts : 

1st.  To  show  that  (l-fa:)"=l-f wic-f,  etc. 
2d.  To  find  the  general  law  of  the  coefiicients. 

First. — To  prove  that  the  coefficient  of  the  second  term 
of  the  expansion  of  {\-\-xY  is  n,  whether  n  be  integral  or 
fractional,  positive  or  negative. 

Let  the  index  be  positive  and  integral ;  then,  since  by  multiplica- 
lion  we  know  that 

(1+^)2=1+2^+-,  etc., 
(\J^xf=l-\-^x-{-,  etc.; 

Let  us  assume  that  {\^xY-'^=\-\-{n — \)x-\-,  etc. 

Multiply  both  sides  of  this  equality  by  1-f-^;  then, 

{l^x)-^l^x)=\l^{n^\)x^,  etc.  }(l-|-a:); 
Or,         {\-\-xY=z\-\-nx-\-,  etc.,  by  multiplication. 


286  RAYS  ALGEBRA,  SECOND  BOOK. 

Hence,  if  the  proposition  is  true  for  any  one  index  n — 1,  it  will 
be  true  for  the  next  higher  index  n.  Now,  by  maltiplication,  it  is 
true  for  the  index  3,  it  is  therefore  true  for  the  index  3-f  1—4;  for 
the  index  4-(-l=5,  and  so  on.  Hence,  by  continued  induction,  it  is 
always  true  for  n  when  it  is  integral  and  positive. 

P 

Next,  let  n  be  a  fraction  =— . 

p 

Also,  let  {l-\-x)q—l-\-ax^,  etc.,   =1-|-Aa;,  where  Ax  is  put  to 

represent  all  the  terms  after  the  first. 
p 
Since  ....     (l-|-a:)?=l+Aa;,  .-.  by  raising  both  sides  to  the 

2' power,  .     .     .     {\-^x)P={l-\-kxf\ 

.-.  l-fpaj+j  etc.,  =l+gAa;+,  etc., 

=l^q{ax-\-,  etc.,)-]-,  etc., 
=l-[-qax^^  etc. 

By  equating  the  coefficients  of  the  like  powers  of  x  (Art.  314), 
p^qa  .-.  aJ^,  and   {l^x)^=\^^x+,  etc. 

Lagtly,  let  n  be  negative;  then,  (Art.  81), 

l\-\-x)-''z=—- — —  =  .= i =l—nx-\-,  etc.,  by  division. 

^    '     '         (l-|-a;)        l-\-nx-\-,  etc.  ^'        '    '' 

Therefore,  {l-{-xy—\-{-nx^,  etc.,  whatever  be  the  value  of  n. 

Therefore,  (a-|-6)''=a"  I  1-j—  V=a«(l_|_72_-|-,  etc.), 

=a'»-{-na"-i6-}-,  etc., 
and  the  first  two  terms  of  the  series  are  determined. 

Second. — To  find  the  general  law  of  the  coeflBcients. 

Let  (l+a-)'*=l-(-r?a;-f  Bx^-f  Ca;3-f  Dx*+,  etc.,  where  B, 
C,  D,  etc.,  depend  upon  n. 

For  Xy  put  x~\-Zy  and  consider  {x-\-Z)  as  one  term;  then, 
|l_f_(a;4-0)  }»=^l4-n(a:+2;)4-B(a:+2;)2+C(a;-}-2:)3-|-,  etc. 
But    .    .    (a-|-6)"=:a"-fna'*-i6-j-,  etc.; 
.-.  {x-^zY=x'^-{-2xz-\-,eic.\ 
{x^z)^=^x^-[-^x'^z-{-,  etc.; 
{x^zy=x^-]-Ax^z-\-,  etc.; 
.-.  {l+(a:-f0)}''==rl-fna:-[-Ba:2-f  Ca:3-1-Dic^+,  etc., 
-f  (n-f-2Ba:-l-3Ca:244Da;3-f ,  etc.)2;-|-,  etc., 
=z(l+a;)'»-|-(n-f 2Ba;-f3Cx2-|-4Da:3-f ,  etc.)0-f-,  etc.,  (A). 


BINOMIAL  THEOREM.  287 

But,  considering  (1+a;)  as  one  term,  {\-\-x^zY=[{l-\-x)-\-z]'^\ 
and  {(l-fa;)-f^]»=iz(l+a:)"+n(l+a:)^-J2;+,  etc.  (B). 

Equating  the  coefficients  of  z  in  (A)  and  (B), 

n-f  2Ba;+3Ca;2-f4Dx3+,  etc.,  =/i(l+a:)"-J. 

Multiplying  both  sides  by  1+iC,  we  have 

n-f  2Ba:+3Cx2+4Da;3-}-,  etc.  -)  :=n(l+a:)'» 
-f  na;4-2Bx2-f3Cic3_(-,  etc.  J  =n(l+na;+Ba;2+Ca;8-|-,  etc.) 

By  equating  the  coefficients  of  the  same  powers  of  ic,  we  have 
2B4-7i=n2  .-.  2B=n2— n=w(n— 1). 
^    n(yi_l) 
^-12' 
3C+2B^Bn  .-.  3C=B(n— 2); 

B(n-2)  _  n(n— l)(n— 2) 
^-      3       ~       1  •  2 • 3      ' 
Also,  4D+3C=nC  .-.  4D=.C(n-3);  * 

_     C(7i— 3)      n(n— l)(n— 2)(n— 3)         ,  .     „  „  ^ 

D^-  -^— ^  =  -5^ ,  ^''^    ^   ^, ; ;  and  so  on  for  E,  F,  G,  etc. 

4  l-2'3-4  '  »'' 

...   (l+.)-^l+n.+"-(»-^l)x3+"("-^f3-^>:^+.  etc.,  and 

.-.  putting  -  for  X,  (a+6)"=a"(  1+-  J, 
h     n(n~-l)b^     n(n—'i){n—2)b^ 

=a-+«a--6+r^a'-^6H"'T^yr^'«-^frH,  etc. 

If  — 6  be  put  for  6,  then  since  the  odd  powers  of  — b  are  negative 
^Art.  193)  and  the  even  powers  positive, 

(«-6)'"=a»-«a"-.6+!i^a-»6^-"i^^<!^a«63+, 

eto.,  which  establishes  the  Binomial  Theorem  ii;  its  most  general 
form.. 

Remark. — ^Prom  the  preceding,  corollaries,  similar  to  those  in 
Art.  310,  may  be  drawn,  but  it  is  not  necessary  to  repeat  them. 
The  following  additional  proposition  is  someti^ucs  useful. 


288  RAY'S  ALGEBRA,  SECOND  BOOK. 

3!S0.  To  find  the  numerically  greatest  term  in  the  expan- 
sion of  {a  -i-  hy. 

I.  If  n  is  a  positive  integer  : 

From  Cor.  3,  Art.  310,  it  appears  that  the  (r-f  1)'*  terra  may  he 

formed   from   the    r«*   by  multiplying  the  latter  by- — .    or 

( 1  \-,    and    this    multiplier    diminishes    as    r    increases; 

while  it  is  greater  than  1,  each  term  is  greater  than  the  preceding; 
and  the  value  of  r,  which  first  makes  it  less  than  1,  indicates  the 
greatest  term ;  that  is,  the  r^^  term  will  be  the  greatest  when 

I  — ■ 1   )-  IS  first  <  1  or  r  first  > —r'-- 

\    r  /a  a-\-b 

From    the    nature    of    the    case,    r   is   necessarily   integral ;    if 

[n-\-Yib  (n-f-l)& 

is  fractional,  take  r  =  the  first  integer  >  — —j-  ,  and  the  r** 

term  will   be  the  greatest.     If  ■    ]"     •   is  an  integer,  and  we  take 


n  will   1 

r=%-   ~\  ,    \  then  the  r"*  term  =  the  (r4-l)'^  and  each  of  these  is 
greater  than    any    other    term ;    for    this    can    only    occur    when 

r      'a 

II.  If  n  is  a  positive  fraction : 

If  —  >  1  there  is  no  greatest  term,  for  the  series  will  evidently 

diverge.     But  if  -  <  1  the  series  will  have  its  greatest  term  (or  terms) 
whose  position  may  be  ascertained  as  in  I. 

III.  If  n  be  negative,  either  an  integer  or  a  fraction  >  1 : 

The  multiplier  that  changes  the  r'*  term  into  the  (r-f-1)'^  viz., 

— ^  .-  may  be  written  —  (  )-i  and  as    the  numerically 

r  a       "^  \      r      /a/ 

greatest  term  is  sought,  disregard  the  sign  of  the  multiplier;  then. 

as  in  I,  the  r'^  term  will  be  the  greatest  when .-  is   first  <  1 

r        a 

„    ,     h{n—Ti) 
or  r  first  >-^ — j-\ 
a — o 

As  in  I,  if J  •  be  a  whole  number  there  are  two  equal  terms 

'        a—b  ^ 

each  greater  than  any  other;   and,  as  in  II,  if  —  be  >  1,   there  is 
no  greatest  term. 


BINOMIAL  THEOREM. 


289 


IV-  If  n  be  negative  and  <  1,  and  -  <  1,  the  first  term  is  the 

greatest;   for  in  this  case  the  multiplier is  <  1  for 

all  values  of  r,  that  is,  each  term  is  less  than  the  preceding. 

Note. — If  b  is  negative,  since  it  is  the  numerical  value  of  the 
term  that  is  to  be  considered,  wfi  mav  disregard  the  sigu  of  b  and 
apply  the  appropriate  one  of  the  preceding  rules.  [Cf.  Todbunter's 
Algebra,  Art.  520.] 

Examples. — Find  the  greatest  term  in  each  of  the  following 
expansions : 

(71+1)6    35 


1.  (2+f)6.     Here 
6.5.4       53     20000 


a+6       11 


1.2.3'^    3«"~ 
2.  (1+|)V. 


81 


3.  (f+lj^ 

4.  (l+i)-i2.    Uere^.^^=^4 


12-13  1        78 
1-2  •52~25' 

5.  (1+f)-^ 


a — b 


Here  ^^^^^^=5 


a—b 


than  any  other  term. 

6.  (l-,V)-i 


r=4  gives  the  greatest  term= 


Ans.  2*^. 
Ans.  5"'  and  6'\ 

-3  gives  the  greatest  term: 


5<Az=6<*,   and   each   is  greater 


Ans.  3'-'^. 


321.   In  the  application  of  the  Binomial  Theorem,  it  is 

merely  necessary  to  take  the  general  formula  (a  +  6)"  —  a'*-|- 

na"~^6+,   etc.,   and    substitute    the    given    quaniities    in    the 

formula,  and  then  reduce  each  term  to  its  most  simple  form 

1 
Example.— 1.  Find  the  expansion  c^f  {l-\-xy. 


..  (x+.)i=l  +  ,.+Mi=Il.^  r^-^^=^^.^+.  etc. 


1     •     1 

T 
2d  Bk. 


1  •  1  •  3 


=1+23^    2  •  4"^^"^2  •  4 
25^^ 


9  •  4 


3  •  5 


6  •  8 


.1-*+,  etc. 


290  RAY'S  ALGEBRA,  SECOND  BOOK. 

Example. — 2.  Develope  (1 — x)~^. 
Here,  a=l,  b=—x,  n=— ^. 

.-.  (l-xf^=l-^{-x)+^—^^\^f±\--xf+,  etc., 

=l+^a:+L|.2+L_|4|,3+,  etc. 

As  the  general   formula  {l-\-x)''—l±:nx-\-   '    ^       x^rb,  etc.,  is 

more  easily  retained  in  the  memory,  and  is  less  complicated,  it  is 
generally  most  convenient  to  reduce  the  quantity  to  be  expanded  to 
this  form.     Thus : 


Develope  ^a-\-b  into  a  series. 

Since  a+b=a(^  ^+|)'  ••  V'^^=v'M  1+-  )^- 


Here,  a;=— ,  n— A  ;  and  since 
(l+.)"=.l+n.4-'^^.^+^^M(  etc., 

••r+a/    -^+^a+  "r^-a2+       1  •  2  •  3      '^3+'  ^^'^ 
b        1    62         1  . 3    63 

—^+2-  "2  .  4a2  +  2  ■  4  •  6'a3    '  ^^''' 

— ,      6       62        53         554 

Hence,    ^a+6=v/a(l+2^-g-2+ ^^3-^23^4+'  ^t^-)- 

1.    ,J^=n.—x)-'^=l-\-x-\-x'-j-x'-^x*Jr,   etc. 
1 — iC 

^    -=(1— a;)-2=l  +  2a:+3x^-f-4rc'4-5a;*-h,  etc. 


,       2x  ,   3x'      4.^  ,   5a;' 


BINOMIAL  THEOREM.  291 


A  oy^i ^         -I  ^^  ^^  5x' 

4.  ^l-.^=l-3-^-gj^-,etc. 


iC  iC'      .         CC' 


5.  ^aHx=a+2^-g-3  +  ^^-j2g^,+,  etc. 

^     ^  ,        ^4  ic  a:^         5a;'  10.x* 

6.  («3_^)^_a-3^,-y^,-g^3-24g^,-,  etc. 

7.  (lH-2a:)-=:l-f-a:— ia:2-[-J-a)3— gx*+,  etc. 

_ —  x^        .T*  a:«  5a;8 

•  ^  la      Ott'      Iba^      lzo«7 

la;      Ix"   .     bx^         10. 


9.  ^a+x=r«(l+37,-9^.+  8n,-54ra,+'  «»«•) 
10.  („,+.3yJ_^i+^^_^^^+_2_5J^^_,  etc.). 

2  1      2   1    .  10  1 


11.  ^9=#'8+l=2+g-g-g-g,  +  gj-g5-,  etc. 

12.  („3_3)._(i_|.  __^^_  ____,  etc.). 

a'  Sx'  ,  2-5x«  ,  2-5-8x'  ,       , 

Here,    —^—-  =  a\a^-x^)-^^=a^X{ci^)^(  ^-fzY^^^'' 


X 


«l-l)-^-«(-l)-*-, 


3SS.  To  find  the  approximate  roots  of  numbers  by  the 
Binomial  Theorem. 

.Let  N  represent  any  proposed  number  whose  n!^  root  is 
required ;  take  a  such  that  a"  is  the  nearest  perfect  n*^  power 


292 

to  N,  so  that  N=a"±&,   h  being  small  compared  with  a, 
and  -j-  or  — ,  according  as  N^  or  <Ca'*; 

Then,  ^N=a  I  l=t:_  )"=:,  by  writing  —    for    h    in    the 

general  formula ; 

,,.16       1    n— 1/  &  \2  ,  1    n— 1  2n— 1/  6  \3 
a{l±-. .-— - 1  —  )  ^-,.  — I         )  _  etc.  . 

Of  this  series  a  few  terms  only,  when  h  is  small  with 
regard  to  a**,  will  give  the  required  root  to  a  considerable 
degree  of  accuracy. 

14.  Required  the  approximate  cube  root  of  128. 


Here,   f  128=f  5H3=5^1+-j-|3; 

^^^3' 125     3'3\125/  ^32*9\125/      '  '  •  *  •  i' 

^^+52—55  ^3-57"  •  •  •  —^+102    m''^rw~  • '  ' 

^5_|_0.04-0.00032+0.0000042—  .  .  . 
=5.0396842. 

323.  In  the  preceding  example,  we  obtain  only  an 
approximate  value.  To  determine  the  limit  in  the  error 
occasioned  by  neglecting  the  remaining  terms  of  the  series, 
let  R  be  the  true  root,  and  as  the  terms  are  alternately 
positive  and  negative,  put 

R  =a — h-\-c — d-{-e—f-\-g — h-\-Jc — 1-{-,  etc.,  and  le* 
R'  =a~h-^c—d-j-e—f, 
'R''=a—h  -f-r— J+e— /-f  y. 

Then,  since  the  terms  continually  decrease,  a — b,  c — d,  e—f, 
g — h,  etc.,  are  all  positive,  and  therefore  R^,  which  contains  three 
only  of  those  diflferences,  will  be  less  than  R.  For  the  same  reason, 
all  the  pairs  of  terms  after  g,  as  — ^-f-^,  — l-\-7n,  etc.,  will  be  all 


BINOMIAL  THEOREM.  293 

negative,  and  W^  will  be  greater  than  R;  therefore,  the  true  value 
of  the  series  lies  between  II"*  and  R^^,  or  between 

a— 6-fc— c?-j-e— /,  and 

a— 64-C— d-|-e— /-|-f/.     Hence, 

The  error  committed  hy  the  omission  of  any  number  of  the 
terms  of  a  converging  series  whose  signs  are  alternately  posi- 
tive and  negative,  is  less  than  the  frst  omitted  term. 

Thus,  in  the  preceding  example,  had  we  stopped  at  the 
3d  term,  the  error  would  have  been  less  than  .0000042. 

15.  Find  the  b'^  root  of  35.  Ans.  2.036172+ 
Here,     .     .     .    N=35=32-|-3=2^(  1+|^  ). 

16.  The  student  may  solve  the  following  examples: 


(1).  i/lO  =|/9-f  1  =:r3.16227  .  .  .  true  to  0.00001. 
(2).  f  24  =fT7^^  =2.88449  .  .  .  true  to  0.00001. 
(3)  v/r08=v/128— 20  ==1.95204  .  .  .  true  to  0.00001. 


Remark. — The  nth  root  may  be  extracted  by  the  formula  in 
Art.  321,  the  number  whose  root  is  to  be  extracted  being  divided 
into  any  two  parts  wliatever.  The  advantage  of  the  formula  in 
Art.  322  consists  in  the  rapid  convergence  of  its  terms. 


THE    DIFFERENTIAL    METHOD    OF    SERIES. 

324.  The  Differential  Method  is  used,  1st,  to  find  the 
successive  differences  of  the  terms  of  a  series ;  2d,  to  find 
any  particular  term  of  the  series ;  or,  3d,  to  find  the  sum 
of  a  finite  number  of  its  terms. 

If,  in  any  series,  we  take  the  first  term  from  the  second, 
the  second  from  the  third,  the  third  from  the  fourth,  and 
so  on,  the  new  series  thus  formed  is  called  the  first  order 
of  differences. 


294  RAY'S  ALGEBRA,  SECOND  BOOK. 

If  we  proceed  with  this  new  series  in  the  same  manner, 
we  shall  obtain  another  series,  termed  the  second  order  of 
differences. 

In  a  similar  manner  we  find  the  third,  fourth,  etc.,  orders 
of  difierences. 

If  we  have  the  series,        1  ,  8  ,  27  ,  64  ,  125  ,  216,     .    .     .     . 
The  1st  order  of  difiFerences  is     7  ,  19  ,  37  ,  Gl  ,  91  ,  .     . 

The  2d       "      "  "  "        12  ,  18  ,  24  ,  30  , 

The  3d      "      "  «  «  ,6,6,6, 


325.  Problem  I. —  To  find  the  first  term  of  any  order 
of  differences. 

Let  the  series  be     a,  h,  c,  d,  e, ;  then,  the 

respective  orders  of  differences  are, 

1st  order,  h — a      ,      c — h      ,      d — c      ,     e — d,  .... 

2d  order,     c—2b-\-a  ,  d—2c-^b  ,  c—2d-^c, 

3d  order,  d — ^3c-f-3^ — a,  e — Sd-\~Sc — h, 

4th  order,  e— 4tZ-|-6c — 4i-[-a. 

Here,  each  difference  pointed  off  by  commas,  though  a  compound 
quantity,  is  called  a  term.  Thus,  the  first  term  in  the  1st  order,  is 
b — a;  in  the  second  order,  c — 2b-{-a,  etc. 

If  we  denote  the  first  terms  in  the  1st,  2d,  3d,  4th,  etc.,  orders  of 
differences  by  Dj,  Dg,  D3,  D4,  etc.,  and  invert  the  order  of  the 
letters,  we  have 

D,=— a+6;  D2=«— 26+e;  Darz^— a+36— 3c-f^d; 
D4=a— 46+6c— 4rf+6,  etc. 

Here,  the  coefficients  of  a,  b,  c,  d,  etc.,  in  the  n'^  order  of  differ- 
ences, are  evidently  those  of  the  terms  of  a  binomial  raised  (0  the 
n'A  power;  and  their  signs  are  alternately  positive  and  negative. 
Hence,  the  first  term  of  the  nth  order  of  differences  is 

a_„6+"_f^e_'^;=l'l!^d+,  etc.,  wl,cn  n  is  .vm,  and 
,      n(n— 1)    ,n(n-l)(n-2),        ,        .  .11 


SERIES— DIFFERENTIAL  METHOD.  295 

Corollary. — It  is  evident  from  the  coefficients  that  when 
n=l,  the  value  of  D^  has  only  two  terms,  for  then  71 — 1=0 ; 
when  n=2,  this  value  has  only  three  terms,  for  then 
71 — 2:=0,  and  so  on. 

1.  Find  the  first  term  of  the  fourth  order  of  difi"erences 
Df  the  series  l^,  2^  3^  4^,  5^  .  .  .  or  1,  8,  27,  64,  etc 

Here,  n=4 ;  hence,  take  five  terms  of  the  first  value  of  D„,  and 
a=l,  b=-8,  0=27,  d=^U,  e==125,  and  I)^= 

1_32+162— 256+125=0,  Ans.  . 

2.  Find  the  first  term  of  the  second  order  of  difi'erences 
of  the  series  1^  2\  3^  4^  .  .  .  or  1,  4,  9,  16,  25. 

Ans.  2. 

3.  What  is  the  first  term  of  the  third  order  of  differ- 
ences of  the  series  1,  3,  6,  10,  15,  etc.?  Ans.  0. 

4.  Required  the  first  term  of  the  fifth  order  of  differ- 
ences of  the  series  1,  3,  3^,  3^,  3*,  etc.  Ans.  32. 

5.  Find  the  first  term  of  the  fifth  order  of  differences 
of  the  series  1,  I,  J,  |,  -^L,  etc.  Ans.  —  3L. 

336.  Problem  II. —  To  find  the  n^^  term  of  the  series 
a,  h,  c,  dy  e,  etc. 

From  the  preceding  article,  we  have  seen  that 

Di= — a-f5;  whence  5=^a-fDl; 

D2==a— 26-fc;  "      c=a+2Di-fD2; 

©3=:— a+-36— 3c-f  d;        "      (irr=a+3Di4  SDs-f-D,; 
D4^a— 46+6c— 4d+e;   «      e=a-f4Di-f  6D2+4D3+D4. 


It  is  evident  from  inspection  that  the  coefficients  of  the  n'^  term 
of  the  series  are  the  coefficients  of  the  (n — 1)  power  of  a  binomial. 


296  RAY'S  ALGEBRA,  SECOND  BOOK. 

Hence,  writing  n — 1  instead  of  71,  in  the  coefficients  of  the  n"*  power 
of  a-f6,  (Art.  319,)  the  nih  term  of  the  series  is 

(n—lMn—^)^      (n— l)(n— 2)(n— 3) 
a+(n-l)Di+  L_Z^ ^  D2+^ 3^-^^-^^ \-}-    etc. 

1.  Find  the  12'^  term  of  the  series  1,  3,  6,  10, 15,  21,  .  . 

1  ,  3  ,  6  ,  10  ,  15, 

2,3,4,5, hence,  Di=^2; 

1,1,1, "      D2-I; 

0,0, "       Do^rO; 

Or,  Di,  D2,  D3,  etc.,  may  be  found  from  the  formula,  (Art.  325,) 
and  the  succeeding  orders  of  differences  are  also  evidently  0;  hence, 
12'/'  term 

^a-Kn-l)D,+("-;»-^^D,=l+llx2+HXL«xl 

=l-f22-f 55=78,  Ans. 

2.  Find  the  n^^  term  of  the  series  2,  6,  12,  20,  30, 
Proceeding  as  above,  to  find  the  orders  of  differences,  we  have 

Di=4,    D2=2,  and  Dg^O; 

hence,  W^  term  =:2-f  (w— 1)4+^ ~\ -V2=n2+n,  Ans. 

From  the  formula  n^-j-n,  or  n(n-\-\)^  any  term  of  this  series  is 
readily  found;  thus,  the  20'^^  term  =20(20+1  )=420. 

It  is  also  evident  that  the  n^^  term  of  a  series  can  be  found  ex- 
actly, only  when  some  order  of  differences  is  zero. 

3.  Find  the  15'^^  term  and  the  71'*  term  of  the  series 
1,  2^  3^  4^  .  .  .  or  l,  4,  9,  le,  .  .  .     Ans.  225,  and  n\ 

4.  Find  the  12'^^  term  of  the  series  1,  5,  15,  35,  70, 
126,  etc.  Ans.  1365. 

5.  Find  the  «'*  term  of  the  series  1,  3,  6,  10,  etc. 


SERIES— DIFFERENTIAL  METHOD.  297 

6.  Find    the    9"^  term  of  the  series    2  •  5  •  7,    4  •  7  •  9, 
6  •  9  •  11,  8    1113,  etc.  Ans.  8694. 

7.  What  is  the  n^^  term  of  the  series  1x2,  3x4,  5x6, 
etc.?  Ans.  4?*^ — 2?t. 

327.  Problem  III.— To  find  the  sum  of  n  terms  of  the 
series  a,   h,  c,  d,  e,  etc. 

Assume  the  series  0,  ot,  a-\~h,  a-j-5-f  c,  a-\-h-{-c-\-d,  .  .  . 

Subtracting  each  term  from  the  next  succeeding,  we  have 

a,  6,  c,  d,  e,  etc., 

which  is  the  series  whose  sum  it  is  proposed  to  find.  Hence,  the 
sum  of  n  terms  of  the  proposed  series,  which  it  is  now  required  to 
find,  is  the  {n-\-iyh  term  of  the  assumed  series. 

It  is  evident  the  y?'A  order  of  differences  in  the  given  series  is 
equal  to  the  (n-(-l)'''  order  in  the  assumed  series.  Hence,  if  we 
compare  the  quantities  in  the  assumed  series,  with  those  of  the 
formula  for  finding  the  n"*  term  of  a  series  (Art.  326),  we  have 

0  for  a,  a  for  Dj, 

n+1  for  n,  Di  for  Do,  etc. 

Substituting  these  values  in  the  formula,  we  have  0-]-{n-\~l — 1)05 
(n+l-l)(n+l-2)^      (n+l-l)(n+l-2)(n+l-3)^ 

+ 1T2 ^^-^ rr^ ^2+, 

,  n(n~l)^   ,  w(n-l)(n-2)^    ,  

Or,   .    .    na-1 — Y-r2~^i+       i  -^-S —    -'^'  ^*^' 

sum  of  n  terms  of  the  proposed  series. 

1,  Find  the  sum  of  n  terms  of  the  odd  numbers  1,  3, 
6,  7,  9,  

Here,  a—1,  D,=2,  D2=0;  hence, 


2.  Find   the  sum  of  n  terms  of  the  series    1^,   2^,  3^ 

4^  5^ 


298  RAY'S  ALGEBRA,  SECOND  BOOK. 

Here,  a^l,  Di^3,  ©2=2,  Dg^O;  hence, 

/i(n-l)(n— 2)_n(n+l)(2n-[-l) 
_f  _  __  _  . 

3.  Find  the  sum  of  7i  terms  of  the  series  l-|-3-|-6+10 

-f.l5,  etc.  «(„-|-l)(.;z+2) 

i\.ns. TT- . 

O 

4.  Find  the  sum  of  20  terms  of  the  series  3+11+31 
+69+131,  etc.  Ans.  44330. 

5.  Find  the  sum  of  20  terms  of  the  series 

1  •  2  •  3+2  •  3  •  4+3  •4-5  +  ,  etc.    Ans.  53130. 

6.  Find  the  sum  of  n  terms  of  the  series  of  cube  num- 
bers 13+23+3^+,  etc.  Ans.  Qw(7i+1)]^ 

7.  Find  the  sum  of  25  terms  of  the  series  whose  n^'^  term 
is  n\Sn—2y  Ans.  305825. 

PILING    OF    CANNON    BALLS    AND    SHELLS. 

328.  Balls  and  shells  are  usually  piled  by  horizontal 
courses,  either  in  the  form  of  a  pyramid  or  a  wedge  ;  the 
base  being  either  an  equilateral  triangle,  or  a  square,  or 
a  rectangle.  In  the  triangle  and  square,  the  pile  termi- 
nates in  a  single  ball,  but  in  the  rectangle  it  finishes  in  a 
ridge,  or  single  row  of  balls. 

3SO*    To  Jinct  the  number  of  halls  in  a  triangular  pile. 

A  triangular  pile,  as  V — ABC,  is  formed  Y 

of  successive  horizontal  courses  of  the 
form  of  an  equilateral  triangle,  the  number 
on  each  side  decreasing  continually  by 
unity  from  the  bottom  to  the  single  ball 
at  the  top. 


SERIES— PILING  OF  BALLS.  299 

If  we  commence  at  the  top,  the  number  of  balls  in  the  respective 
courses  will  be  as  follows: 

1st,  2<i.  3d.  4<\  5«A. 


•••• 

••••• 

•• 

••• 

••• 

•••• 

• 

•• 

•• 

••• 

• 

• 

•• 

and  so  on.     ITcnce,  the  number  of  balls  in  the  respective  courses 

is  1,    1+2,   l+2-f3,   1+2+3+4,    1+2+3+4+5,   and  so  on;   or, 
13  6  10  15 

Hence,  to  find  the  number  of  balls  in  a  triangular  pile,  is  to  find 
the  sum  of  the  series  1,  3,  6,  10,  15,  etc.,  to  as  many  terms  (n)  as 
there  are  balls  in  one  side  of  the  lowest  course. 

By  applying  the  formula  (Art.  327)  to  finding  the  sum  of  n  terms 
of  the  series  1,  3,  6,  10,  etc.,  we  have  a=l,  Di=:2,  D2=:l,  and 
I>3=0. 

Hence,  the  formula  na-\ — \^ — ^I^iH — —^i — jr^q — ^2  gives 

n^—Sn^-\-2n     n(n+l)(n+2) 
n+n2_n+ g— — =-i_X.^A_!U.        (A) 

330*    To  find  the  number  of  halls  in  a  square  pile. 

A  square  pile,  as  V — EFH,  is  formed  V 

of  successive  square  horizontal  courses, 
such  that  the  number  of  balls  in  the 
sides  of  these  courses  decreases  con- 
tinually by  unity,  from  the  bottom  to 
the  single  ball  at  the  top. 

If  we  commence  at  the  top,  the  number  of  balls  in  the  respective 
courses  will  be  as  follows: 

1»«.         2<^.         3'^.         4<A. 

•••      •••• 

••       •••      •••• 

•        ••       •••      •••• 

•  ••• 


300  RAY'S  ALGEBRA,  SECOND  BOOK. 

and  so  on.  Rence,  the  number  of  balls  in  the  respective  courses 
is  12,  22,  32,  42,  52,  etc.,  or  1,  4,  9,  16,  25,  and  so  on.  Therefore, 
to  find  the  number  of  balls  in  a  square  pile,  is  to  find  the  sum  of 
the  squares  of  1,  2,  3,  etc.,  to  as  many  terms  (n)  as  there  are  balla 
in  one  side  of  the  lowest  course. 

But  this  sura  (Ex.  2,  pp.  297,  298)  is  ^(^+^H-^+^),         (g) 


331.   To  find  the  number  of  halls  in  a  rectangular  pile, 

A  rectangular  pile,  as  EFD 
BCA,  is  formed  of  successive 
rectangular  courses,  the  num- 
ber of  balls  in  each  of  the 
sides  decreasing  by  unity  from 
the  bottom  to  the  single  row 
at  the  top. 


a.a.jLJV»-J^^A^**-'»^^i 


If  we  commence  at  the  top,  the  number  of  balls  in  the  breadth 
of  the  successive  rows  is  1,  2,  3,  and  so  on.  Also,  if  m-fl  denotes 
the  number  of  balls  in  the  top  row,  the  number  in  the  length  of  the 
second  row  will  be  m-[-2,  in  the  third,  w-f  3,  and  so  on.  Hence,  the 
number  in  the  respective  courses,  commencing  with  the  top,  will 
be  l(m+l),  2(771+2),  3(7n+3),  and  in  the  n<A  course  n{m-\-n). 
Or, 

S=l(m+l)+2(?n4-2)4-3(?n-f3)+ _^n(m+n) 

=m(l-f2+3+4    .    .    .    -]-n)+(12+22+32-f-42+     .    .     -fn2); 

but  the  sum  of  n  terms  of  the  series  in  the  two  parentheses  (Arts. 

327.330,)is»-i!^',and"'"+^f"+^l     Hence, 

Jt  DO 

Here,  m-f-n  represents  the  number  of  balls  in  the  length  of  the 
lowest  course.  If  we  put  r7i-f-n=^,  we  have  Zr)i-\-2n=Zl—n;  sub- 
stituting this  for  3m-|-2n,  in  (C),  we  have 


SERIES— PILING  OF  BALLS.  301 

It  is  evident  that  the  number  of  courses  in  a  triangular  or  square 
pile  is  equal  to  the  number  of  balls  in  one  side  of  the  base  course, 
and  in  the  rectangular  pile  to  the  number  of  balls  in  the  breadth  of 
the  base  course. 

33S(.  Collecting  together  the  results  of  the  three  pre- 
ceding articles,  we  have  for  the  number  of  balls  in  a 

Triangular  pile      -n(n+l)(7i+2)      ....       (A); 
Square  pile  -^n{n-\-l){2n-\-\)  ....       (B); 

Rectangular  pile   ^n(w-j-l)(3? — ^*-j-l)    •     •     •       (C). 

In  (A)  and  (B),  n  denotes  the  number  of  courses,  or 
number  of  balls  in  the  base  course.  In  (C),  n  denotes  the 
number  in  the  breadth,  and  I  the  number  in  the  length, 
of  the  base  course. 

The  number  of  balls  in  an  incomplete  pile  is  evidently 
found  by  subtracting  the  number  in  the  pile  which  is 
wanting  at  the  top,  from  the  whole  pile  considered  as  com- 
plete. 

1.  Find  the  number  of  balls  in  a  triangular  pile  of  15 
courses.  Ans.   680. 

Here,  n=15.     Substituting  this  value  in  (A),  we  find  the  number 

J5(15+1H15+2)^15X16X17^6S0    ^„, 
2X3  6  ' 

2.  Find  the  number  of  balls  in  an  incomplete  triangular 
pile  of  15  courses,  having  21  balls  in  the  upper  course. 

From  the  illustrations  in  Art.  329,  it  is  evident  that  the  number 
of  balls  in  one  side  of  the  upper  course  is  6;  therefore,  5  courses 
have  been  removed  from  the  pile.  From  formula  (A),  we  find  that 
the  pile,  considered  as  complete,  would  contain  1540  balls,  and  that 
the  removed  pile  contains  35.  Hence,  1540—35^1505,  the  number 
left. 


302  RAY'S  ALGEBRA,  SECOND  BOOK. 

3.  Find  the  number  of  balls  in  a  square  pile  of  15 
courses.  Ans.  1240. 

4.  Find  the  number  of  balls  in  a  rectangular  pile,  the 
length  and  breadth  of  the  base  containing  52  and  34  balls 
respectively.  Ans.  24395. 

5.  Find  the  numbier  of  balls  in  an  incomplete  triangular 
pile,  a  side  of  the  base  course  having  25  balls,  and  a  side 
of  the  top  13.  Ans.  2561. 

6.  How  many  balls  in  an  incomplete  triangular  pile  of 
15  courses,  having  38  balls  in  a  side  of  the  base? 

Ans.  ^7580. 

7.  Find  the  number  of  balls  in  an  incomplete  square 
pile,  a  side  of  the  base  course  having  44  balls,  and  a  side 
of  the  top  22.  Ans.  26059. 

8.  The  whole  number  of  balls  in  the  base  and  top  courses 
of  a  square  pile  are  1521  and  169  respectively;  how  many 
are  in  the  incomplete  pile?  Ans.  19890. 

9.  The  number  of  balls  in  a  complete  rectangular  pile 
of  20  courses  is  6440  ;   how  many  balls  are  in  its  base? 

Ans.  740. 

10.  The  number  of  balls  in  a  triangular  pile  is  to  the 
number  in  a  square  pile  having  the  same  number  of  balls 
in  the  side  of  the  base  as  6  to  11  ;  required  the  number 
in  each  pile.  Ans.  816,  and  1496. 

11.  How  many  balls  are  in  an  incomplete  rectangular 
pile  of  8  courses,  having  36  balls  in  the  longer  side,  and 
17  in  the  shorter  side  of  the  upper  course?    Ans.  6520. 

INTERPOLATION    OF    SERIES. 

333.  Interpolation  is  the  process  of  finding  interme- 
diate numbers  in  mathematical,  astronomical,  or  other 
tables.  Its  object  is  to  furnish  a  shorter  method  of  com- 
pleting such  tables  when  portions  of  them  have  been  cal- 
culated by  formulae. 


INTERPOLATION  OF  SERIES.  303 

Thus,  if  the  logarithms  of  5,  6,  and  8,  are  respectively 
0.6989,  0.7782,  and  0.9031,  it  may  be  required  from 
these  data  to  find  the  logarithm  of  7. 

The  latter  numbers  are  sometimes  called  functions  of 
the  former,  and  the  Ibrmer  arguments  of  the  functions. 

As  the  functions  constitute  a  series,  the  principle  upon 
which  interpolation  is  founded  is  explained  in  Art.  326; 
that  is,  certain  terms  of  a  series  being  known,  it  is  re- 
quired to  find  the  ?i'^  term. 

Three  cases  may  arise,  which  we  will  now  consider. 

Case  I. — When  the  differences  of  the  functions  are  pro- 
portional, or  nearly  proportional,  to  the  differences  of  the 
arguments^  or  the  functions  are  in  arithmetical  progres- 
sion. 

Ex. — Given  the  Dip  of  the  Sea  Horizon  at  the  heights 
of  86,  89,  92,  95,  and  98  feet,  viz.,  9'08",  9'17", 
9'26",  9'36",  and  9'45";  required  that  of  101  ieet. 

Ans.  9'54". 

Here,  the  first  differences  being  9^^,  or  nearly  so,  we  add  9^^  to 
9^45^^  for  the  Dip  at  lOI  feet. 

In  all  practical  examples,  there  is  no  common  first  difference, 
and  it  becomes  necessary  to  employ  the  second,  third,  etc.,  differ- 
ences. If  in  the  series  composing  the  functions,  we  can  obtain  an 
order  of  diflFerences  equal  to  zero,  the  interpolation  will  be  exact. 
In  most  cases,  however,  D2,  D3,  etc.,  do  not  vanish,  but  become  so 
small  after  D2  or  D3  that  they  may  be  omitted  without  sensible 
error. 

334.  Case  II. — When  the  differences  of  the  functions 
are  not  proportional  to  the  differences  of  the  arguments, 
and  the  term  to  he  interpolated  is  one  of  the  equidistant 
functions. 

Ex.— Given  ^"2^=2  92401 ,  |^26=2  96249,  f'2Y=S, 
^29=3.07231,  to  find  the  cube  root  of  28. 


304        RAY  S  ALGEBRA,  SECOND  BOOK. 

lu  such  examples,  when  three  quantities  are  given,  we  may  sup- 
pose D3  to  vanish  or  become  very  small.  We  then  have  (Art.  326) 
the  equation  — a-(-36— 3c-f  (ir=0,  and  any  of  the  quantities  a,  6,  c, 
or  d,  may  be  found,  when  the  other  three  are  given.  Similarly,  if 
the  fourth  differences  vanish,  then 

a— 46+6c— 4rf+e=0. 

In  the  above  example,  four  quantities  are  given  to  find  a  fifth; 
therefore,  we  have  a — 46-f6c — 4<i-|-e=^0,  where  d  is  the  term  to  be 
interpolated;  hence,  4d:i=a-f 6c -fe  — 46  =2.92401 -f  18+3.07231 
^11.84996=12.14636,  where  d,  or  f  "28=3.03659,  which  is  true 
to  .00001. 

335.  Case  III. — "When  the  differences  are  as  in  Case 
2d,  and  the  term  to  be  interpolated  is  intermediate  to  any 
two  of  the  functions. 

Ex.— Having  given  the  logarithms  of  102,  103,  104, 
and  105,  let  it  be  required  to  find  the  logarithm  of 
103.55. 

Taking  the  formula,  Art.  326,  put  p  to  represent  the  distance,  in 
intervals^  of  the  required  term  [t)  from  a,  the  first  term  of  the  series, 
in  which  case  p=n — 1,  since  the  number  of  intervals  is  one  less  than 
the  number  of  terms.     Then, 

<^„+pD.+^ifdlD,+^'^-'>(>;-^)D3+,  etc. 

The  intervals  between  the  given  numbers  is  always  to  be  consid- 
ered as  unity,  and  p  is  to  be  reckoned  in  parts  of  this  interval; 
hence,  p  will  be  fractional. 

Sufficient  accui\acy  is  generally  obtained  by  making  use  of  Dj  and 
D2  only,  in  the  above  formula. 

Ill  practice,  however,  the  following  is  generally  adopted: 

Take  the  two  functions  of  the  series  which  precede,  and  the  two 
which  follow  the  term  required,  and  find  from  them  the  three  first 
differences,  and  the  two  second  differences.  Call  the  second  of  the 
three  first  differences  d,  the  mean  of  the  two  second  differences  d\ 
the  fractional  part  of  the  interval  p',  and  second  term  6.  We  then 
have  from  the  above  formula, 

t^bJ,p\d^-^d'), 


INFINITE  SERIES. 


305 


Applying  this  formula  to  the  above  example,  we  have 


Nos. 

Logarithms. 

1st  Diff. 

2d  Diff. 

Mean  of 
2d  DilT. 

102 
103 
104 
105 

2.0086002 
2.0128372 
2.0170333 
2.0211893 

42370 
41961 
41560 

-409 
-401 

—405 

Here,  iy-=.55,  rf=41961,  d^=— 405,  and  6=2.0128372. 
^=2.0128372  +.55(41961+^X405). 
<=2.0128372+.0023129=^2.0151501,  Ans. 

1.  Find  the  2*^  term  of  the  series  of  which  the  4'^*  dif- 
ferences vanish,  the  1*',  3'^,  4'^  and  b^^  terms  being  3,  15, 
30,  55;  and  find  the  6^  7^  and  8"^  terms. 

Ans.  7;   and  93,  147,  and  220. 

2.  Find  the  5'^  term  of  the  series  of  which  the  6"^  dif- 
ferences vanish,  and  the  1*',  2^^,  3*^,  4'^,  6'^  and  7'*  terms 
are  11,  18,  30,  50,  132,  209.  Ans.  82. 

3.  Given  the  logarithms  of  101,  102,  104,  and  105; 
viz.,  2.0043214,  2.0086002,  2.0170333,  and  2.0211893, 
to  find  the  logarithm  of  103.  Ans.  2.0128372. 

4.  Given  the  cube  roots  of  60,  62,  64,  and  66;  viz., 
3.91487,  3.95789,  4,  and  4.04124,  to  find  the  cube  root 
of  63.  Ans.  3.97905. 

5.  Having  given  the  squares  of  any  two  consecutive 
whole  numbers,  show  how  the  squares  of  the  succeeding 
whole  numbers  may  be  obtained  by  addition. 


INFINITE    SERIES. 

336.  An  Infinite  Series  is  a  series  consisting  of  an 
unlimited  number  of  terms. 

The  Sum  of  an  infinite  series  is  the  limit  to  which  we 
approach  by  adding  together  more  terms,  but  which  can 
2d  Bk.  26 


306  RAY  S  ALGEBRA,  SECOND  BOOK. 

not  be  exceeded  by  adding  together  any  number  of  terms 
whatever. 

A  Convergent  Series  is  one  which  has  a  sum  or  limit. 
Thus,  l+i+i  +  .+Jj+3'3+gL+,  etc., 

is  a  convergent  series,  whose  limit  is  2,  since  the  sum  of 
any  number  of  its  terms  can  not  exceed  2. 

A  Divergent  Series  is  one  which  has  no  sum  or  limit;  as, 

l_^2-f4+8+16+32+,  etc. 

An  Ascending  Series  is  one  in  which  the  powers  of  the 
leading  quantity  continually  increase  ;  as, 

a-\-hx-\-cx^-\-ddi?-\-. 

A  Descending  Series  is  one  in  which  the  powers  of  the 
leading  quantity  continually  diminish  ;  as, 

'  '  '  '  '  ^  X      x^      x^  ^ 

S37.  There  are  four  general  methods  of  converting  an 
algebraic  expression  into  an  infinite  series  of  equivalent 
value,  each  of  which  has  been  already  exemplified ;  viz., 

1st.  By  Division^  Art.  134;  2d.  By  Extraction  of  Roots, 
Art.  183;  3d.  By  Indeterminate  Coefficients,  Arts.  315-7; 
and,  4th,  By  the  Binomial  Theorem,  Art.  321. 

338.  The  Summation  of  a  Series  is  the  finding  a 
finite  expression  equivalent  to  the  series. 

The  General  Term  of  a  Series  is  an  expression  from 
which  the  several  terms  of  the  series  may  be  derived  ac- 
cording to  some  determinate  law. 

Thus,  in  the  series  :p-)- 5-  + K--|-;r+ the  general 

a,  1      J     o      4 

term  is  —    because  by  making  X—\,  2,  3,  etc.,  each  term  of  the 

X 
series  is  found. 

Again,  in  the  series  2  •  2-f 2  •  3-f2  •  4+2  -5+ the 

general  term  is  2(a;-|-l). 


INFINITE  SERIES.  307 

As  different  series  are  in  general  governed  by  different 
laws,  the  methods  of  finding  the  sum,  which  are  applicable 
to  one  class,  will  not  apply  universally. 

We  present  two  methods  of  most  general  application. 

First  Method. — In   a   regular  decreasing  geometrical 

series,  whose  first  term  is  a,  and  ratio  r,  the  sum  is  = 

(Art.  299). 

Second  Method. — By  subtraction. 

Ex.  1. — Find  the  sum  of  the  infinite  series  ^^ — r>-|-o — i 
I  ^  2  •  8  '  3  •  4 

+  4^5  +  5^-^'^'^- 

Then,  |  +  l+i+ J-f,  etc.,  =  S-J. 

Subtracting  273  + 3T4  + 4-75  + 5T6+'  ^^c.,  =  ^,  Ans. 

Ex.  2. — Find  the  sum  of  the  infinite  series 


J        I  1   3 '  a-  5 

+  6^  +  7^+''''-  .     • 

Then,  i+i+i+^+  etc.,  :r^  S  -1. 

Subtracting  j"^  +  3^  +  5^+,  etc,  =  1,  and  ^-L  +  ^^ 

In  such  series,  the  first  factor  in  the  successive  denominators  is 
variable,  while  the  second  factor  exceeds  the  first  by  a  constant 

quantity.     The  general  term  is  therefore     ,     ,  where  n  is  vari- 

^  ->  ^  n{n-\-py 

able  and  p  constant. 
Since  1^^=-P?^^  ...         '^       =l{g__^|. 

From  which  we  derive  the  foUowinGr 


308  RAY'S  ALGEBRA,  SECOND  BOOK. 

Rule. — Having    found  the  values  of  q,  n,  and  p,  in  the 
given  series,  express   the    series   whose   general  formulas  are 

-  and  — ^ — ;  subtract  the  latter  from  the  former,  and  divide 
n  n-\-p'  J  J  ^ 

the  result  hy  p  for  the  sum  of  the  series. 

1.  Required  the  sum  of  the  series  ■= — ^-f"  6 — s:  +  ''= — tt+j 
etc.,  ad  infinitum,  that  is,  to  infinity. 
Here,    .     .     .     q=l,  p=2,  and  n=A,  3,  5,  7,  etc. 

Then,   .     .     .     ^=1+1+1+4+,  etc,  ad  inf. 
Subtracting, ^—  =  -~-^ — r=l ; 


n{n+py 


:^z^  sum  of  given  series. 


The  sum  of  n  terms  of  the  same  series  is  found  in  a  manner  nearly 
similar.     Thus, 

l=^+m+^ d-i 

^^=i+5+h 2^i^  +  2n+i 

<? <J     ^     P<J     ^l L_=J^and— i— =33-^   Ans 

n     n-{-p     n{n-\-p)  2n-|-l      2n-f-l        n{n-^p)    2n-}-V 

2.  Find  the  sum  of  the  series  -rT-o  +  oTq  H~  qT1.H~'®*^  » 
ad  infinitum. 

Here,  ^=1,  p=lj  and  w=l,  2,  3,  etc.  Ans.  1. 

3.  Find  the  sum  of  the  above  series  to  n  terms. 

Ans.  — -=-. 

n-|-l 


RECURRING  SERIES.  309 

4.  Find  the  sum  of  tlie  series  r^ — -^  4-  ^ — ^  -f-  ■^^—^■4-  -^ — =,. 

1  -4^2  •5^3-6^4  •  7 
-f-,  etc.,  ad  infinitum. 

Here,  2=1,  and  p=S,  7i=l,  2,  3,  etc.         Ans.  j|. 

5.  Find  the  sum  of  the  series  = — ^  -f  ^y—-.  -f  k— ? -}-,  etc., 
ad  infinitum. 

Here,  ^=1,  p=2,  and  n=l,  2,  3,  etc.  Ans.  |. 

6.  Find  the  series  whose  creneral  term  is  j-  ;  also 

w(n-{-4) 
find  its  sum  continued  to  infinity. 

AQ-  1,1,1.1.  25 

A.  Series  =^-^  +  ^-^  +  ^  _|.  ^_^+,  etc.,  sum  =^. 

The  sums  of  series  may  often  be  found  by  reducing  them,  by 
multiplication  or  division,  to  the  forms  already  known.     Thus, 

7.  Find  the  sum  of  the  series  l+i  +  e-fTo-f  ?  ^^^-i  ^^ 
infinitum.  Ans.  2. 

Divide  by  2  and  compare  with  example  2d. 

8.  Find  the  sum  of  the  series  n— q  +  ts — To-f  n — Ti'H"? 

o  *  o      u  ■  IIZ      y '  lb 

etc.,  ad  infinitum.     (Multiply  by  3  •  4).  Ans.  J,,. 

Remark. — There  are  other  methods  for  the  summation  of  cer- 
tain classes  of  series,  but  they  are  too  complex  and  extensive  for 
an  elementary  work. 


RECURRING    SERIES. 

3S9.  A  Recurring  Series  is  a  series  so  constituted  that 
every  term  is  connected  with  one  or  more  of  the  terms 
which  precede  it  by  an  invariable  law,  usually  dependent 
on  the  operations  of  addition,  subtraction,  etc. 

Thus,  in  the  series  l-\-2x4-3x'2^5x^^8x*-\-13x^Jr^lx^^,  etc., 
the  sum  of  the  coefficients  of  any  two  consecutive  terms  is  equal 


310 

to  the  coefficient  of  the  next  following  term;  and,  by  means  of 
this  relation  between  the  coefficients,  the  series  may  be  extended 
to  any  desired  number  of  terms. 


340.  The  particular  relation,  by  means  of  which  the 
coefficient  of  any  term  of  the  series  may  be  found  when 
the  preceding  coefficients  are  known,  is  called  the  scale  of 
tJie  coefficients.  It  is  easily  seen  that  it  is  sufficient  to  find 
the  successive  co'efficie7its  in  order  to  determine  the  series, 
inasmuch  as  the  desired  powers  of  the  variable  may  be 
supplied  as  wanted. 

Recurring  series  are  of  the  fi^^st  order,  second  order,  etc., 
according  to  the  number  of  terms  in  the  scale. 

b       h^        b^ 
Thus,  in  the  series  1 x-\ — ^x^ gX^  etc.,  the  coefficient  of 

each  term  after  the  first  is  equal  to  the  preceding  coefficient  multi- 
plied by ,  and  the  series  is  said  to  be  of  the  first  order.     This, 

the  simplest  form  of  a  recurring  series,  is  obviously  a  series  in 
Geometrical  Prosrression. 


341.    To  find  the  scale  of  the  coefficients  of  a  recurring 

series. 

When  the  series  is  of  the  first  order,  the  scale  is  easily 
determined,  being  the  ratio  of  any  two  consecutive  coef- 
ficients.    (Art.  295.) 

When  the  series  is  of  the  second  order,  the  law  of  the 
series  depends  on  two  terms,  and  the  scale  consists  of  two 
parts. 

Let  p  and  q  represent  the  two  terms  of  the  scale  of  the 
coefficients  of  the  recurring  series, 

A+Ba;+Ca:2+Dar'4-Ea;^+JV,  etc., 


RECURRING  SERIES.  311 

Then,  by  the  assumed  law  of  the  series : 

C=Bp+Aq',  (1) 

D-Cp+Bg;  (2) 

E=DpfCg;  etc.    (3) 

The  values  of  p  and  q  may  be  found  by  eliminating  between  any 
two  of  these  equations.     Taking  the  first  two,  (Art.  158.) 

BC-AT)  ,        BD— C2 

•••    P=W^:^AC    ^"^^=F=AC 

Ex.— Find  the  scale  of  the  series  l~^2x-{-3x'-{-4:(^-^5x\ 
etc. 

Here,  A-^l,  B=2,  C=3,  D=4,  etc. 

•••    P=  2^-1X3  ^^    ^"<^5=2^=Iix3=-l- 

Now,  by  the  use  of  the  scale,  we  may  extend  the  series  as  far  as 
we  please :  the  5th  coefficient  =jpX  the  4thH-gX  the  3d=2X4— 3 
=5;  the  6th  coefficient=2X5— 4^6;  the  7th=^2X6— 5^7,  and  as 
the  ascending  powers  of  x  are  wanted,  we  have  Qx^  for  the  6th 
term,  7x^  for  the  7th,  etc. 


S4!3.  In  a  recurring  series  of  the  third  order,  the  law 
of  the  series  involves  three  terms,  which  we  will  represent 
by  py  q,  and  r,  the  series  being  A-\-'Bx~\-Cx'^-{-Dx^-{-Kx^-\- 
Fa^-\-Go^,  etc. 

Then,  by  the  law  of  the  series, 

I>:^Cp^Bq\-Ar; 
E:-Dp+Cg+Br; 
F=Ep-\-Dq+Cr',  etc., 

And,  by  combining  these  equations,  the  values  of  p,  q,  and  r  are 
readily  found,  (Art.  158.)  In  a  similar  manner  the  scale  may  be 
determined  in  series  of  the  higher  orders. 

In  finding  the  scale  of  a  series,  we  must  first  ascertain 
by  inspection  whether  the  series  is  in  G.  P. ;  if  not,  then 


312  RAY'S  ALGEBRA,  SECOND  BOOK. 

make  trial  of  a  scale  containing  two  terms,  then  one  of 
three,  four,  and  so  on,  until  a  correct  result  is  obtained. 
We  must  be  careful  not  to  assume  too  many  terms ;  for  in 
that  case  every  term  of  the  scale  will  take  the  form  ^. 

343.  To  find  the  sum  of  an  infinite  recurring  series  whose 
scale  of  relation  is  known. 

Let  A-I-Bx+Cx^+Da^'-J-Ex*,  etc.,  be  a  recurring  series 
of  the  second  order,  p  and  q  being  the  terms  of  the  scale. 

Then,     .     .     .     A=A; 

Ca:2=Bpa:2-f-Aga;2; 
T)0(?=Cpx-'-[-Bqx^;  etc.,  ad  infinitum. 

Represent  by  S  the  required  sum,  and  add  together  the  corre- 
sponding members  of  the  preceding  equations,  observing  that  Ba;-|- 
Cx'^-\-Y)x^-\-f  etc.,  =S — A  ;  then,  we  have 

S=A+Brc+ (S— A)  pa^-f  Sga;2 ; 
.  • .    S— Spa;— Sgx2= A+Ba:— Apa; ; 

Or,  ...    .    S=^±J?^=^. 
'  1 — px—qx^ 

If  we  make  q^=^,  (remembering  that  B=Ap),  the  formula  be- 

comes  8=:; ,  which  is,  as  it  ought  to  be,  identical  with  the 

l—px'  '  *  ' 

formula  of  Art  299. 

Remark. — Every  recurring  series  may  be  supposed  to  arise  from 
the  development  of  a  rational  fraction,  and  the  summation  of  such 
a  series  may  be  regarded  as  a  return  to  the  generating  fraction. 
There  are  several  methods  of  accomplishing  this  return :  of  these 
the  preceding  is  regarded  as  the  most  suitable  for  an  elementary 
work. 

1.  Find  the  sum  of  l  +  Sx+Sa^^+Tx-^+Px*,  etc. 

Here,  A=l,  B=3,  C=5,  ^=7,  etc. 
And,  hence,  (Art.  341.)  p=2,  (/=— 1. 

A  \-Bx~kpx_\ -f-3a;— 2t;_  1  -\-x 
men,  fe-  ■^_^^,_^^^2  -i_2.T+a;2-(l-a;)2* 


REVERSION  OF  SERIES.  313 

In  each  of  the  following  series,  find  the  scale  of  rela- 
tion, and  the  sum  (S)  of  an  infinite  number  of  terms  : 

2.  l-j-6x-\rl2x'Jr^8x'-i-120x'-\-,  etc. 

3.  l-]-2x-^Sx'-\-^x^-^bx'-\-ex'-\-,  etc. 

Ans.  p=2,   q=-l;    S==-J_. 

4. ^H -, — 1-,  etc. 

Ans.     The  series  is  iu  G.  P.  p= ;  S= — '■ — 

^        ,  c  '         c-j-bx 

5.  x-\-x^-\-3^-\-,  etc. 

Alls.  The  series  is  in  G.  P.  p=l ;  S^    ^ 

1 — X 

6.  X — x^-\~x^ — x*-\-,  etc. 

Ans.  The  series  is  in  G.  P.  p=—l ;  S=— ^. 

7.  l  +  3a:+5a;2+7x3+9a;*+,  etc. 
Ans.  p=2,  q=—l,  S:  ■^^'^ 


8.  l^+2^a;+3V+4V+5V+6V4-,  etc. 

Ans.  p=d,  q=~S,  r=l  ;    S.r= J-t^ 


■(1-x) 


REVERSION    OF    SERIES. 


344.  To  Revert  a  Series  is  to  express  the  value  of 
the  unknown  quantity  in  it  by  means  of  another  series  in- 
volving the  powers  of  some  other  quantity. 

Let  X  and  y  represent  two  undetermined  quantities,  and 
express  the  value  of  y  by  a  series  involving  the  powers 
of  x ;  thus, 

7/=ax-\-bx'^-\-coi^-\-dx*-{-,  etc.,  (1), 

in  which  a,   b,  c,   d,  etc.,  are  known  quantities;  then,  to 
revert  this  series  is  to  express  the  value  of  x  in  a  series 
2d  Bk.        27* 


314  RAY'S  ALGEBRA,  SECOND  BOOK. 

containing  the  known  quantities  a,  6,  c,  d,  etc.,  and  the 
powers  of  y. 

To  revert  this  series,  assume  x=Ay-]-By^-{-Cy^-]-T)y^,  etc.  (2),  in 
which  the  coefi&cients  A,  B,  C     .     .     .     aie  undetermined. 
Eind  the  values  of  y^,  y^,  y*     .     .     .     from  (1),  thus, 

y-=a^x^-\-2abx^-\-{b^-{-2ac)x*-{-.    .... 
2/3=  a^x^-{-Sa-bx^-{-    .... 

y*=  d^x^-^    ....  etc. 

Substituting  these  values  in  (2),  and  arranging,  we  have- 

a;4-f ,  etc. 


0=Aa 

rc+A6    a;24-     Ac 

ic3+      kd 

-1 

Ba2        -f2Ba6 

+      B62 

+    Ca3 

4-  2Bac 

+  3Ca26 

+     Da4 

Since  this  is  true,  whatever  be  the  value  of  x^  and  the  coefficients 
of  iC,  a:2,  x^^  etc.,  will  each  =0,  (Art.  314,  Cor.),  we  have 

Aa-1 =0,  .-.  A=-, 

a' 


A6+Ba2 =0,  .-.  B=-A 


Ac+2Ba6+Ca3 ==0,  .-.  d 


262_ac 


Ad-fBd2_|_2Bac+3Ca26+Da<=0,  .-.  d=_'L'^'~^^^+^^ 


a7 


1          6   „    262_ac  „     a2d-5a6c-}~563  . 
Hence,  a:=- ?/-  -gg/H      ^5     ^ ^7 ^  +,  etc.  (3) 


345.  If  the  given  series  has  a  constant  term  prefixed, 
thus,  y=^a!-\-ax-{-hj?-\-c7?-\-dx^-{- 

assume  y — a!=.z.^  and  we  have 

z=zax-\-hxP--\-c^-\-dx^-\-^  etc. 

But  this  is  the  same  as  (1)  in  the  preceding  article,  except  that 
z  stands  in  the  place  of  y\  hence,   if  2  be  substituted  for  y  in 


REVERSION  OF  SERIES.  315 

(3,)   [Art.  344],  the  result  will  be  the  required  development  of  X] 
and  then,  y — a'  being  substituted  for  Z,  the  result  is 

2  f)  21)2 (2c 

^=-(2/-«0-^(2/-«0'+  —^^{y-ct'r-   etc. 


346.  When  the  given  series  contains  the  odd  powers 
of  X,  assume  for  x  another  series  containing  the  odd  powers 
of  y.     Thus,  if    y=zcex-{-hx^-\-cx''-\-dx''-{- 

to  develope  x  in  terms  of  ^,  assume 

x=A^+Bf-\-C7/-{-J)i/'+ 

Then,  by  substituting  the  values  of  y,  2/^,  etc.,  derived  from  the 
former  equation,  in  the  latter,  and  equating  the  coefficients  to  zero, 
we  find 

1  6        Sb^—ac  ,     a^d—8abc+12b^  ^ 

If  both  sides  of  the  equation  be  expressed  in  a  series,  as 

^2/H-^2/'+C2/^+j  etc.,  =a^x-\-b^x^^&x^-{-,  etc., 

and  it  be  required  to  find  y  in  terms  of  X,  we  must  assume,  as 
before, 

y=zAx^Bx^-\~Cx^-^Dx*-{-,  etc., 

and  substitute  the  values  of  y,  y^,  y^,  etc.,  derived  from  this  last 
equation,  in  the  proposed  equation;  we  shall  then,  by  equating  the 
coefficients  of  the  like  powers  of  x,  determine  the  values  of  A,  B,  C, 
etc.,  as  before. 

The  following  exercises  may  be  solved  either  by  substi- 
tuting the  values  of  a,  5,  r,  etc.,  in  the  equations  obtained 
in  the  preceding  articles,  or  by  proceeding  according  to 
the  methods  by  which  those  equations  were  obtained. 

1.  Given  the  series  i/=x — x'^-\-x^ — x^-\~  ....  to  find  the 
value  of  X  in  terms  of  y.       Ans.  x=i/-\-y'^-^y^-\-y^-\-,  etc. 

Find  the  value  of  a*,  in  an  infinite  series  in  terms  of  y: 

2.  AVhen  i/=x-\-x'^-{-oc^-\-,  etc. 

Ans.  x=y — y'^~\-y^ — y*'~\~'!^ — )  ®^^' 


316 

3.  When  7/=2x-\-Sx'^4:a^-\-5x-'-]-,  etc. 

Ans.  ^-c^^y— /gy+i'/g/— ,  etc. 

4.  When  3/=:l— 2a:+3x^ 

Ans.  a:=-J.(y-l)  +  3(3,_l)^__9_(^__l)3+,  etc. 

5.  When  y=a:+^.x2+Ja;3_|__^i_^4_|_^  el-^, 

Ans.  cc^:^— ^/+J/— 1/4-,  etc. 

6.  When  y+^y+^y+^y  •  •  •  =gx-\-hx'^-\-kx^-\-lx*'  .  .  . 
Ans  x=^   I  (^^ -^y  I  W-^9-^K^f~h)y 


XI.     CONTINUED     FRACTIONS :     LOGARITHMS : 

EXPONENTIAL    EQUATIONS:    INTEREST, 

AND    ANNUITIES. 

CONTINUED    FRACTIONS. 

34T.  A  Continued  Fraction  is  one  whose  denomina- 
tor is  continued  by  being  itself  a  mixed  nuinher,  and  the 
denominator  of  the  fractional  part  again  continued  as  be- 
fore, and  so  on ;  thus, 


1 

1 

1 

-r 

"+1 

in  which  a,  5,  c,  c?,  etc.,  are  positive  whole  numbers. 

Continued  fractions  are  useful  in  approximating  to  the 
values  of  ratios  expressed  by  large  numbers,  in  resolving 
exponential  equations,  indeterminate  equations  of  the  first 
degree,  etc. 


CONTINUED  FRACTIONS.  317 

34:8«    To  express  a  rational  fraction  in  the  form  of  a 

continued  fraction. 

80 
Let  it  be  required  to  reduce  y^m  ^^  ^  continued  fraction. 

If  we  divide  both  terms  of  the  fraction  by  the  numerator,  we 

^+30 

7  1 

If  we  omit  ^n,  the  denominator  will  be  too  small  and  =,  the  value 

6\j  5 

of  the  fraction,  will  be  too  large. 

7 
Again,  if  we  divide  both  terms  of  the  fraction  ^  by  the  numer- 

ator,  we  find  r-,^=^ = . 

'  157      ^     1 

2                                                      1  4 

If  we  omit  =,  the  value  will  be  expressed  by :,  =oT»  which  is 

less  than  the  true  value  of  the  fraction.     Hence,  generally, 

By  stopping  at  an  odd  reduction^  and  neglecting  the  frac- 
tional part,  the  result  is  too  great ;  hut  hy  stopping  at  an 
even  reduction,  and  neglecting  the  fractional  part,  the  result 
is  too  small. 

Since  -= =^,  we  find 

so     1 

=-c^=- T  1»<  reduction,  too  great; 

'       5-1 2d  "  too  small; 

4-\-- 3^    .       «         too  great; 

3-1-  4^^         "         true  value. 

By  this  process  we  find 

13      1 49       1 

30  =  ^^1  204-^^1 

3+5  6+g 


318  RAY'S  ALGEBRA,  SECOND  ]500K. 

349«  The  different  quantities 
111 


a+-        a-\ ^^  etc., 

are  called  converging  fractions,  because  each  one  in  succes- 
sion gives  a  nearer  value  of  the  given  expression.     -* 

The  fractions  — ,  ^,  — ,  etc.,  are  called  integral  fractions. 
a    b   c^        ^  ^ 


330.    To  explain   the  manner  in  which  the  converging 
fractions  are  found  from  the  integral  fractions. 

1.    - :=—  1*'  conv.  fraction. 

a  a 


2.        ,  1  =— — — f   2'' conv.  fraction. 

a+^ ab-\-l 

1 


a-\- zi     ,     ,     ,      =  ^   ■■  ,  ^,   , —   S'^  conv.  fraction. 

1  c(aZ;+l)-j-a 

c 


By  examining  the  third  converging  fraction,  we  find  it  is  formed 
from  the  1'',  and  2<^,  and  from  the  3^  integral  fraction,  as  follows: 

Num.     =3^^  quot.Xnum.  of  2<^  conv.  fract.-|-num.  of  1*'  conv.  fract. 
Denom.=::3<^  quot.Xden.    of  2'^  conv.  fract.-|-den.    of  l"'  conv.  fract. 

P     Q    R 

To  prove  the  general  law  of  formation,  let  p^  ^T/j  p/  he  the  three 

converging  fractions  corresponding  to  the  three  integral  fractious 
— ,  V,  and  -,  and,  as  has  already  been  shown, 

R  _Q  c+P 


CONTINUED  FRACTIONS.  319 

1  S 

Let  us  now  take  the  next  integral  fraction  -r,  and  let  -^  express 

R 
the  4'A  converging  fraction.     Then,  it  is  obvious  that  z^  will  become 

S  1 

^  by  substituting  c-f--^,  instead  of  C;  hence, 

S^_   Q  [  <^+a  /  "^  ^  _  (Q  c+P  )d+Q  _  Rc^-fQ 


Q^(o4)+P 


From  this  we  see  that  the  same  rule  applies  to  the  4'^  converging 
fraction,  and  so  on.     Hence,  for  the  n'^  converging  fraction, 

Multiple/  the  denominator  of  the  n*^  integral  fraction  hy 
the  numerator  of  the  (n — 1)'*  converging  fraction^  and  add 
to  the  product  the  numerator  of  the  (n — 2)"^  converging 
fraction.  This  will  give  the  numerator  of  the  d'*  converging 
fraction. 

Multiply  the  denominator  of  the  n^^  integral  fraction  hy 
the  denominator  of  the  (a — 1)"*  converging  fraction^  and 
add  to  the  product  the  denominator  of  the  (n — 2)'^*  converg- 
ing fraction.  This  will  give  the  denominator  of  the  n''*  con- 
verging fraction. 

Ex. — To  find  a  series  of  converging  fractions  for  ^^^ly. 

The  integral  fractions  are  ^,  |,  i,  |,  |,  i,  ^. 

The  converging  fractions  are  1,  -J,  |,  /y,  ^  j%\,  ^^^. 

351.  If  the  2^  converging  fraction  (Art.  350)  be  sub- 
tracted from  the  1*',  the  remainder  will  be  found  to  be  a 
fraction  having  for  its  numerator  +1,  and  for  its  denom- 
inator the  product  of  the  two  denominators  ;  and  if  the 
3*^  be  subtracted  from  the  2'^,  the  resulting  fraction  will 
have  — 1  for  its  numerator,  and  the  product  of  the  denom- 
inntors  for  its  denominator. 


320  RAY'S  ALGEBRA,  SECOND  BOOK. 

By  a  process  of  reasoning  similar  to  that  employed  in 
Art.  850,  it  may  be  shown,  in  a  general  manner,  that 

T^e  difference  between  any  two  consecutive  converging  frac- 
tions is  always  a  fraction  having  -^-1,  or  — 1,  for  its  numer- 
ator, and  the  product  of  the  two  denominators  for  its  denom- 
inator, according  as  the  fraction  subtracted  is  in  an  even  or 
odd  place. 

S52m    To   show  that  every  converging  fraction  is  in  its 

lowest  terms;  and  to  find  the  approximate  value  of  the  frac- 

a 

tion  Y' 

o 

A  C 

If  —  and  —   be   any  two  consecutive   converging  fractions,  by 

Art.  351,  |_5:3.+2^,  or  -^;  that  is,  AD-BC=±1. 

Now,  if  A  and  B  have  a  common  divisor  greater  than  1,  it  will 
divide  their  multiples  AD  and  BC,  and  their  difference  rirl,  (Art, 
100);  or,  a  quantity  greater  than  1  is  a  divisor  of  1,  which  is  im- 

possible;    hence,  —  is  in  its  lowest  terms. 

Since  the  denominators  of  the  convergents  continually  increase, 
and  their  values  continually  diminish,  and  since  the  true  value  of 

J  lies  between  any  two  consecutive  convergents,  it  is  evident  that 

by  continuing  the  series,  any  degree  of  approximation  to  the  true 
value  may  be  obtained. 

3»>3.  To  express  |/N,  t^Ae?i  N=a'+1,  in  the  form  of 
a  continued  fraction. 

1 

^a^4.1_a+^/a2^1_a^a+-;==^  (Art.  206), 


2a-fj^ — |-,  etc. 


Ex.  -/l7=|/4^-f-l=4 


CONTINUED  FRACTIONS. 
1 


321 


8-j-T^-|-,  etc.,  the  converg- 

.       ^       .  1.       11  .        A  1     8     65 

ing  tractions  to  be  added  to  4,  are  q,  ^,  ^oo5  ^^^• 


3S4.   To   convert   y^N,  where  Nrnia^-j-^j   i^^o    a   con- 
tinued fraction. 


Ex.  Convert  |/19,  or  |/16-|-3,  into  a  continued  frac- 
tion. 

1 


Hence, 


1/19=4+-. 

1       _,/T9+4  1 

«-^/19_4 3—''+  5- 


Hence, 


_^_  _3(/19-h2)_^19+2_^  ,  1 
~^19— 2~         15        ~       5      ~^G 

~^19-3~         10        ~       2  "*"d- 

•     •     v/19^4+^ 

1+^+,  etc. 


Proceeding  in  the  same  manner,  the  successive  values  of  a,  6,  c, 
cZ,  e,  and  /  will  be  found  2,  1,  3,  1,  2,  8.  The  value  of  g  is  the  same 
as  that  of  a  ;  consequently,  the  succeeding  values  will  recur  in  the 
same  order  as  before. 

The  converging  fractions  are  4,  S,    L3,  IS^   61^  etc. 


33S.  To  find  the  value  of  a  continued  fraction^  when  the 
denominators  q,  r,  s,  etc.,  of  the  integral  fractions  recur  ad 
infinitum  in  a  certain  order. 


322  RAY'S  ALGEBRA,  SECOND  BOOK. 

1 


Ex.  1.— Let 


.+'- 


,  1 

r 


^_j_z,  etc.,  ad  infinitum. 


Then,     ....    ^  =x,    or    — ~ — r^=^- 

From  this  equation,  the  value  of  x  is  easily  found. 

3S6.    To  find  in  the  form  of  a  continued  fraction  the 
value  of  X,  which  satisfies  the  eqiiation  a^=b. 

Ex. — Required  the  value  of  a;  in  the  equation  10-^^2. 

By  substituting  0  and  1  for  a:,  it  appears  that  a:>0  and  <\, 

1  1 

Let    .     .      x=--,  then,  10'/=2,  or  2^=10. 

y 

Since  23=8,   and   2''=16,  one  of  which   is   less  and  the  otlier 

greater  than  10 ;  therefore,  2/>3,  and  <4;    let  2/=3-| — ; 

z 

Then, 2H^=10; 

1  \ 

Or, 2^.2^=10,  or  2^  =  lyO^l. 25; 

Therefore, (1.25)''=2. 

Again,  it   appears    that    2;>3,    and    <4;    let  0=3-^ — ;  then, 

(  1.25  )3+«  =(1. 25)3(1. 2.5)«  =2  .-.  (1.25)«z=  ^-^^1.024; 

Therefore,  (1.024)"=:1.25,  and  by  trial  «>9  and  <10. 
1_ 

3+: 


Hence, a;_ 


3+9+,  etc. 
This  gives  a:=J—    y3g+,  || -=^.30107  nearly,  etc. 


CONTINUED  FRACTIONS.  323 

Reduce  each  of  the  following  to  a  continued  fraction,  and 
find  the  successive  integral  and  converging  fractions : 


1. 

130 

Ans. 

Integral  fractions 

1 

'3' 

h 

hh- 

421 

Converging     " 

h 

1% 

-  y.  ^1?- 

2. 

130 

Ans, 

Integral  fractions 

h 

h 

hk- 

291 

Converging     " 

h 

i, 

ih  m- 

3. 

157 

Ans. 

Integral  fractions 

h 

h 

h  i,  h 

972 

Converging     " 

1 
6' 

A 

2  1         6  8       1  o  7 
5    1305    43iT'    975 

4.  The  height  of  Mt.  Etna  is  10963  feet,  of  Vesuvius 
3900  feet;  required  the  approximate  ratio  of  the  height 
of  the  former  to  that  of  the  latter. 

Ans      '       J      -S-     ifi        37         90       127       3900 
-^"o-    3'    3'     14'    45'    T04'   253»    ~367»    10963* 

5.  The  height  of  Mt.  Perdu,  the  highest  of  the  Pyrenees, 
is  11283  feet;  that  of  Mt.  Hecla  is  4900  feet;  required 
the  approximate  ratio  of  the  height  of  the  former  to  that 
of  the  latter.  Ans.  i,  |,  ig^  ||,  -^^3,  etc. 

6.  When  the  diameter  of  a  circle  is  1,  the  circumfer- 
ence is  found  to  be  greater  than  3.1415926,  and  less 
than  3.1415927  ;  required  the  series  of  fractions  converg- 
ing to  the  ratio  of  the  circumference  to  the  diameter. 

Ans.  ^,  3^,  ^o«,  andilf. 

Show  that  this  last  ratio,  J  if,  is  true  to  within  less  than 
three  ten-millionths  of  the  circumference. 

Suggestion.— In  examples  of  this  kind  the  integral  fractions, 
corresponding  to  both  fractions,  should  be  found,  and  then  the  con- 
verging fractions  calculated  from  those  integral  fractions  that  are 
the  same  in  both  series. 

7.  Express  approximately  the  ratio  of  24  hr.  to  5  hr., 
48  min.,  49  sec,  the  excess  of  the  solar  yr.  above  365  da. 

An«      1        "7         8         31  39         655         694        1349      20  92  9 

Hence,  after  every  4  years,  we  must  have  had  1  intercalary  day, 
as  in  leap  year;  after  every  29  years,  we  ought  to  have  had  7  inter- 
calary days;  after  every  33  years  we  ought  to  have  had  8  inter- 


324  RAY'S  ALGEBRA,  SECOND  BOOK. 

calary  days.  This  last  was  the  correction  used  by  the  Persian 
astronomers,  who  had  7  regular  leap  years,  and  then  deferred  the 
eightli  until  the  fifth  year,  instead  of  having  it  on  the  fourth. 

8.  Find  the  least  fraction  with  only  two  figures  in  each 
term,  approximating  to  l^H-  Ans.  |A. 

9.  The  lunar  month,  calculated  on  an  average  of  100 
years,  is  27.321661  days.  Find  a  series  of  common  frac- 
tions approximating  nearer  and  nearer  to  this  quantity. 

Ans      27      82      765      3907      pfp 

^ns.  -J-,  -3-,  -2g-,  -y4  3  ,  eic. 

10.  Find  a  series  effractions  converging  to  y2. 

A  n «      1       3      7      17      4  1      p+p 

11.  Show  that  y5  is  >  |g|,  and  <  ff|9. 

12.  If  8-==32,  find  a: ^   ^    Ans.  s. 

13.  If  3-=15,  find  a; Ans.  2.465. 

LOGARITHMS 

3S7*  This  method  of  computation  was  invented  by 
Jjonl  Napier,  but  subsequently  much  improved  by  Mr. 
Henri/  Briggs,  whose  system  is  now  universally  adopted  in 
numerical  computations. 

The  advantage,  secured  in  the  use  of  logarithms,  arises 
from  the  application  of  the  law  of  exponents,  by  which 
multiplication,  division,  involution,  and  evolution  are  per- 
formed by  addition,  subtraction,  multiplication,  and  divi- 
sion. 

cii  

Thus,  a^X  «'=«',     -3=«*,   {ay=a^\  fd'^=.a^. 

If  some  number,  arbitrarily  assumed,  be  taken  as  a  hase, 
then 

The  Logarithm  of  any  mimher  is  the  exponent  of  that 
power  of  the  base,  which  is  equal  to  that  number. 

Thus,  if  a  is  the  base  of  a  system  of  logarithms,  N,  N'',  N'''',  etc., 
any  numbers,  and 


LOGARITHMS.  325 

a2=^N,  a^-^^W,  a^=N^^, 

then,  2,  3,  and  X  are  called  the  logarithms  of  N,  N'',  and  N^^,  in 
the  system  whose  base  is  a. 

The  base  of  "  Briggs'  Logarithms,"  or  the  common  system,  is  the 
number  10,     Assuming  this,  we  shall  have 


(10)0=1 

(10)1=10 

(10)2=100 

(10)3=1000 

(10)4=10000 


hence,  0  is  the  log.  of  1 ; 
"   1  "  »  log.  of  10; 
"   2  "  "  log.  of  100; 
"   3  "  "  log.  of  1000 ; 
"   4  "  "  log.  of  10000; 


Etc.,  Etc. 

The  logarithm  of  every  number  between  1  and  10  is,  evidently, 
some  number  between  0  and  1 ;   that  is,  a  proper  fraction. 

The  logarithm  of  every  number  between  10  and  100  is  some  num- 
ber between  1  and  2:   that  is,  1  plus  a  fraction. 

The  logarithm  of  every  number  between  100  and  1000  is  2  plus  a 
fraction  ;  and  so  on. 

SSS.  The  integral  part  of  a  logarithm  is  called  the  in- 
dex or  characteristic  of  the  logarithm. 

Since  the  logarithm  of  1  is  0,  of  10  is  1,  of  100  is  2, 
»  of  1000  is  3,  and  so  on  ;  therefore,  for  any  number  greater 
than  unity, 

The  Characteristic  of  the  logarithm  is  one  less  than  the 
number  of  integral  figures  in  the  given  number. 

Thus,  the  logarithm  of  123  is  2  plus  a  fraction  ;  the 
logarithm  of  1234  is  3  plus  a  fraction,  and  so  on. 

339.  The  computation  of  logarithms,  in  the  common 
system,  consists  in  finding  the  values  of  x  in  the  equation 

10"^=N,  when  N  is  successively  1,  2,  3,  etc. 

One  method  of  finding  an  approximate  value  of  x  has 
been  explained  in  Art.  356,  but  other  methods  more  ex- 
peditious will  be  given  hereafter. 


326 


RAYS  ALGEBRA,  SECOND  BOOK. 


The  following  table  contains  the  logarithms  of  numbers 


from  1  to  100  in  the  common  system  : 


N. 

Log. 

N. 

Log. 

N. 

Lug. 

N. 
76 

Log. 

1 

0.000000 

26 

1.414973 

51 

1.707570 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158 

63 

1.724276 

78 

1.892096 

4 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

0.698970 

30 
31 

1.477121 

55 

1.740363 

80 

1.903090 

6 

0.778151 

1.491362 

56 

1.748188 

81 

1.908486 

7 

0.845098 

32 

1.505150 

57 

1.766876 

82 

1.913814 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479 

69 

1.770852 

84 

1.924279 

10 

1.000000 

1 

35 
36 

1.544068 

60 

1.778151 

85 

1.929419 

n 

1.041393 

1.556303 

61 

1.785330 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939619 

13 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128  1 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 

1.176091 

40 

1.60206.) 

65 

1.812913 

90 
91 

1.964243 

16 

1.204120 

41 

1.612784 

66 

1.819544 

1.969041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832609 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 

1.301030 

45 

1.653213 

70 

1.845098 

96 

1.977724 

21 

1.322219 

46 

1.662758 

71 

1.851258 

96 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.867333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1.991226 

24 

1.380211 

49 

1.690196  ' 

74 

1.869232 

99 

1.995635 

25 

1.397940 

50 

1.698970 

75 

1.875061 

100 

2.000000 

In  works  on  Trigonometry,  Surveying,  etc.,  where  a  set  of  loga- 
rithmic tables  is  given,  the  characteristic  is  usually  omitted,  and  must 
be  supplied  by  the  rule  given  in  Art.  358. 


360.  General  Properties  of  Logarithms. — Let  N  and 

N'  be  any  two  numbers,  x  and  x'  their  respective  loga- 
rithms, and  a  the  base  of  the  system  ;  or,  take  any  two 
numbers  in  the  common  system.     Then,  (Art.  357), 


lO^r-100000, 
10^^100, 


(1), 


LOOAKITHMS.  327 

Multiplying  equations  (1)  and  (2)  together,  we  find 

10' =10000000, r=a^+*'=NN^. 

But,  by  the  definition  of  logarithms,  7  and  x^x^  are  the  loga- 
rithms of  10000000  and  of  NN^  respectively.     Hence, 

TliG  sum  of  the  logarithms  of  tico  numhers  is  equal  to  the 
logarithm  of  their  product. 

Similarly,  the  sum  of  the  logarithms  of  three  or  more 
factors,  is  equal  to  the  logarithm  of  their  product.  Hence, 
to  multiply  two  or  more  numbers  by  means  of  logarithms. 

Rule. — Add  together  the  logarithms  of  the  numhers  for 
the  logarithm  of  the  product. 

361.  Taking  the  same  equations,  (Art.  360),  we  have 

10^^:100000,  a-=N   ....  (1), 

10-^^=100,  a^=W  ....  (2). 

Dividing  equation  (1)  by  equation  (2),  we  find 
103=1000,     ....    a^^=f-. 

But,  by  the  definition  of  logarithms,  3  and  X—x^  are  the  loga- 

N 
rithms  of  1000  and  of  ^.    Hence,  to  divide  by  means  of  logarithms, 

Kule. — From  the  logarithm  of  the  dividend  suhfract  the 
logarithm  of  the  divisor  for  the  logarithm  of  the  quotient. 

1.  Find  the  product  of  9  and  6  by  means  of  logarithms. 

♦ 

By  the  table  (page  326),  the  log.  of  9  is     .     .     .     .     0.954243 

"         "  the  log.  of  6  is     .     .     .     .     0.778151 

The  sum  of  these  logarithms  is 1.732394 

and  the  number  corresponding  in  the  table  is  54, 

2.  Find  the  quotient  of  63  divided  by  9,  by  means  of 
logarithms. 


328  RAY'S  ALGEBRA,  SECOND  BOOK. 

The  log,  of  63  is 1.799341 

'•    log.  of    9  is 0.954243 

The  difference  is 0.845098 

and  the  number  corresponding  to  this  log.  is  7. 

By  means  of  logarithms 

8.  Find  the  product  of  7  and  8. 

4.  Find  the  continued  product  of  2,  3,  and  7. 

5.  Find  the  quotient  of  85  divided  by  17. 

6.  Find  the  quotient  of  91  divided  by  13. 

36!S.  Resuming  equation  (1),  (Art.  360),  we  have 

10'=:100, a"=N. 

Raising  both  sides  to  the  3d  and  to  the  rrV-^  power,  we  find 

lGfi=1000000, a'^=N^ 

But,  (Art.  357),  6  and  mx  are  the  logarithms  of  1000000  and 
of  N"*  respectively  Hence,  to  raise  a  number  to  any  power  by 
nicans  of  logarithms, 

Hule. — Multiply  the  logarithm  of  the  given  number  by  the 
expoyient  of  the  required  power  for  the  logarithm  of  the 
power  of  the  number. 

363.  Take  the  same  equation 

10^=1000000, a'=N. 

Extracting  the  3d  and  n<^  root  of  both  sides,  we  have 

«     102:^100, a":=N". 

X  - 

But,  (Art.  357),  2  and  —  are  the  logarithms  of  100  and  of  N" 

respectively.     Hence,  to  extract  any  root  of  a  number. 

Rule. — Divide  the  logarithm  of  the  given  member  by  the 
index  of  the  required  root  for  the  logarithm  of  the  root  of  the 
number. 


LOGARITHMS,  329 

1.  Find  the  third  power  of  4  by  means  of  logarithms. 

The  logarithm  of  4  is 0.602060 

Multiply  by  the  exponent  3 3 

The  product  is 1.806180 

which  is  the  logarithm  of  64. 

2.  Extract  the  fifth  root  of  32  by  means  of  logarithms. 

The  logarithm  of  32  is 1.505150 

Dividing  by  the  index  5,  the  quotient  is  ...     .      0.301030 
which  is  the  logarithm  of  2,  the  required  root.    ^ 

Solve  the  following  examples  by  means  of  logarithms : 

3.  Find  the  square  of  7. 

4.  Find  the  fourth  power  of  3. 

5.  Extract  the  cube  root  of  27. 

6.  Extract  the  sixth  root  of  64. 

Other  examples  may  be  taken  from  arithmetic.  It  is,  however, 
the  province  of  algebra  to  explain  the  principles  of  logarithms,  and 
the  methods  of  computing  the  tables,  rather  than  their  use  in  actual 
calculations. 

364.  By  means  of  negative  exponents,  we  can  also  ex- 
press the  logarithms  of  fractions  less  than  1.  Thus,  in  the 
common  system,  since 

(10)-^=-',       =.1       ,  therefore,  —1  is  the  log.  of  .1 
(10)-2=_ij(j     =.01     ,         "         —2      "      log.  of  .01 
(10)-3=_^i^_  =.001  ,         "         —3      "      log.  of  .001 

(^^)-'=l^hm=-^^^         "  -4      "      log- of  -0001: 

Etc.,    '  Etc. 

The  logarithm  of  any  fraction  between  one  and  one-tenth,  between 
one-tenth  and  one-hundredth,  etc.,  may  be  expressed  thus, 

Log.  (f^)==  log.  (JoX7)=:  log.  J^+  log.  7—1+  log.  7. 
I-og-  (t!u)=  l^g-  (fkX3)-  log.  -fio+  l«g-  3=-2+  log.  3. 
2d  Bk.  28 


330  RAY'S  ALGEBRA,  SECOND  BOOK. 

It  is  customary  not  to  perform  the  subtraction  indicated,  but  to 
unite  the  logarithm  of  the  numerator  to  the  negative  characteristic. 
Thus, 

Log.  0.7  =—1+  log.  7=— 1.845098,  or  1.845098. 
Log.  0.03  =—2+  log.  3=— 2.477121,  or  2.477121. 
Log.  0.004=— 3+  log.  4=— 3.602060,  or  3.602060. 

Hence,  the  characteristic  of  the  logarithm  of  a  decimal 
fraction  is  a  negative  number^  and  is  numerically  equal  to 
the  distance  of  the  first  significant  figure  from  the  decimal 
point. 

36S.  On  the  piinciple  above  explained,  we  may  deduce 
the  following 

General  Rule  for  finding  from  the  Tables  the  Loga- 
rithms of  any  Decimal  Fraction.—!.  Find  the  logarithm 
of  the  figures  composing  the  decimal  as  if  the  fraction  were 
a  whole  number. 

2.  Prefix  the  negative  characteristic  according  to  the  rule 
given  in  Art.  364. 

360.  The  following  examples,  illustrative  of  the  prin- 
ciples already  explained,  will  afford  a  useful  exercise : 

1.  Log.  (a.b  .c.d.  .  )=  log.  a-\-  log.  b-\-  log.  c-j-  log.  d. 

2.  Log.  I  -y-  1=  log.  a-f-  log.  b-\-  log.  c —  log.  d —  log.  e. 

3.  Log.  (cC^  .h^  .  c^  .  )=m  log.  a-f  71  log.  b-\-2^  log.  c. 

— ^ —  I  z=m  log.  a-]-n  log.  b — p  log.  c 

5.  Log.  (a^ — x'^)=  log.  [(a-}-^)(« — ^)]=  log-  («-f^)  + 
log.  (a—x). 


6.  Log.  |/a''' — x^=}i  log.  (a-{-x)-\-^  log.  (a — a-). 


LOGARITHMS.  331 

7.  Log.  (a^Xif^^Sf  log.  a. 
8-  Log.  ^^^^Wllog.  (a-a.)-3  log.  (a-fc.)  j. 

367.  Let  us  resume  the  equation  a^=N. 

1st.  If  we  make  x=l,  we  have  a^=:N=a ;  hence,  log. 
.=1 ;  that  is, 

TTAa^ever  he  the  base  of  the  system,  its  logarithm  in  that 
system  is  1. 

2d.  If  we  make  x=0,  in  the  equation  a*==N,  we  have 

a^=l^—-.l ;  hence,  log.  1=0  ;  that  is, 
In  any  system  the  logarithm  of  1  is  0. 

368.  In  the  equation  a=^=N,  consider  a>l,  as  in  the 
common  and  the  Naperian  systems,  and  x  negative  ;  we 
then  have 

a-^=i=N,  and  ^=a^  =0,  or  log.  0=— oo  . 
I*  a 

Hence,  the  logarithm  of  0,  in  a  system  whose  hase  is  greater 
than  1,  is  an  infinite  number  and  negative. 

In  a  similar  manner,  it  may  be  shown  that  in  a  system  whose 
base  is  less  than  1,  the  logarithm  of  0  is  infinite  and  positive. 

369.  As  the  positive  and  negative  characteristics  are 
taken  to  designate  whole  numbers  and  fractions,  there  re- 
mains no  method  of  designating  negative  quantities  by 
logarithms ;  or,  as  N,  in  each  of  the  equations  a*=N  and 
a~*=N,  is  positive, 

Negative  numbers  have  no  real  logarithms. 


332 


COMPUTATION    OF    LOGARITHMS. 

370.  Before  proceeding  to  explain  the  methods  of 
computing  logarithms,  we  may  observe  that  it  is  only  neces- 
sary to  compute  the  logarithms  of  the  prime-  numbers. 

For,  the  logarithm  of  every  composite  number  is  equal  to  the  sum 
of  the  logarithms  of  its  factors.  Hence,  the  logarithms  of  1,  2,  8, 
6.  7,  etc.,  being  known,  we  can  find  those  of  4,  6,  8,  etc.     Thus, 


4=22 

hence 

log. 

4=2  log.  2,  (Art.  362); 

6=2X3 

(( 

log. 

6=    log.  2-j-  log.  3; 

8=23      . 

(( 

log. 

8=3  log.  2; 

9=32 

(( 

log. 

9=2  log.  3; 

10=2X5 

(( 

log. 

10=    log.  2+  log.  5. 

1.  Suppose  the  logarithms  of  the  numbers  2,  3,  5,  and 
7  to  be  known  ;  show  how  the  logarithms  of  the  compo- 
site numbers  from  12  to  30  may  be  found. 

2.  Of  what  numbers  between  30  and  100,  may  the  loga- 
rithms be  found  from  those  of  2,  3,  5,  and  Y;  and  why? 

Ans.  Of  23  diflferent  numbers,  from  32  to  98. 

371*  In  the  common  system,  the  equation  a^=N  (Art. 
357)  becomes  10"=N. 

If  we  multiply  both  sides  by  10,  we  have 

10'XlO=10^+^=10N; 
Also,    .    .    .    10^X100=10^X10^=10'^^=100N. 

Hence,  in  the  common  system,  the  logarithm  of  any  number  will 
become  the  logarithm  of  10  times,  100  times,  etc.,  that  number,  by 
increasing  the  characteristic  by  1,  2  etc.  From  this  results  the 
advantage  of  Briggi  system. 

Thus,  the  log.  of  3      is 0.477121, 

«  «  30     " 1.477121, 

«  "  300   « 2.477121. 


LOGARITHMS.  333 

Also,  the  log.  of      .2583  is —1.412124, 

2.583    " 0.412124, 

«  25.83      " 1.412124. 

37S*  If  we  compare  the  different  powers  of  10  with 
their  logarithms  in  the  common  system,  we  have 

Numbers       1  ,     10  ,     100,  1000,  10000, 
Logarithms  0,1,        2,3,        4      ,  and  so  on. 

Hence,  while  the  Jiumbers  are  in  geometrical  progression^ 
their  logarithms  are  in  arithmetical  progression. 

Therefore,  if  we  take  a  geometrical  mean  between  two 
numbers,  and  an  arithmetical  mean  between  their  loga- 
rithms, the  latter  number  will  be  the  logarithm  of  the 
former. 

Thus,  tlft  geometrical  mean  between  10  and  1000  is  |/I0xl000 
=100,  and  the  arithmetical  mean  between  their  logarithms,  1  and  3, 
is  (l-|-3)--2=2. 

In  general,  if  a*=N,  and  a^':z=W]  then, 

Log.  of  y'Niris^i^. 

By  means  of  this  principle,  the  common,  or  Briggean,  system  of 
logarithms  was  originally  calculated. 

Ex. — Let  it  be  required  to  calculate  the  logarithm  of  5. 

First. — The  proposed  number  lies  between  1  and  10;  hence,  its 
logarithm  will  lie  between  0  and  1. 

The  geometrical  mean  is  i/(lXl0)=3.162277;  the  arithmetical 
mean  is  (0-f  l)-=-2=0.5.     Hence,  the  log.  of  3.162277  is  0.5. 

Secondly. — Take  the  numbers  3.162277  and  10,  and  their  loga- 
rithms .5  and  1,  we  find 
The  log  of  5.623413  is  0.75. 

Thirdly.— T&kQ  the  numbers  3.162277  and  5.623413,   and  their 
logarithms  0.5  and  0.75,  we  find 
The  log.  of  4.216964  is  0.625. 


^34  RAY'S  ALGEBRA,  SECOND  BOOK. 

Fourthly.— Take  the  numbers  4.216964  and  5.623413,  and  their 
logarithms  0.625  and  0.75,  we  find 

The  log.  of  4.869674  is  0.6875. 

By  continuing  this  process,  always  taking  the  two  numbers  near- 
est to  5,  one  of  which  is  less  and  the  other  greater,  after  twenty-two 
operations,  we  obtain  the  number  5.000000-(-,  and  its  corresponding 
logarithm  0.698970+. 

Having  the  log.  of  5  we  readily  find  that  of  2,  or  '^  (Art.  361). 

To  find  the  log.  of  3,  take  the  numbers  2  and  3.162277,  and  their 
logarithms,  and  proceed  as  in  finding  the  log.  of  5. 


37S.  Logarithmic  Series.— The  most  convenient  method 
of  computing  logarithms  is  by  means  of  Series,  which  we 
shall  now  proceed  to  explain. 

Let  ic  be  a  number  whose  logarithm  is  to  be  expressed 
in  a  series,  and  let  us  apply  the  method  of  Indoierminato 
Coefficients  (Art.  314). 

If  we  assume  log.  x—A-\-Bx^Cx'^-{-Dx^-j-,  etc.,  and  make  x=0, 
we  have,  log.  0=A=qo  (Art.  368).     Hence, 

00  =:A,  which  is  absurd. 

If  we  assume  log.  Xz=A.x-{-Bx^~\-Cx^-\-,  etc.,  and  make  a;=rO,  we 
have  log.  0=:^0;  that  is,  (Art.  368),  qo  =^0,  which  is  also  absurd. 
Hence,  it  is  impossible  to  dcvelope  the  logarithm  of  a  number  in 
powers  of  that  number. 

But  if  we  assume 

Log.  {l-\-x)=Ax^Bx-~\-Cx^'-{-T)x*-}-,  etc.     .     .     (1) 

and  make  X:^0,  we  have  log.  1=:0,  which  is  correct  (Art.  367). 

In  like  manner,  also  assume 

Log.  {l-\-z)=Az-\-Bz^-{-Cz^--{-'Dz*-}-,  etc.    ...     (2) 
Subtracting  equation  (2)  from  (1)  we  get 

Log.  (l-frc)— log.  {l-\-z)=A(x—z)-^B{x^—z^) 
-f  C(a;^— 2;3)+,  etc.    .    .     .     (3). 


LOGARITHMS.  335 

The  second  member  of  this  equation  is  divisible  by  x — z  Art.  83); 
let  us  reduce  the  first  member  to  a  form  in  which  it  shall  also  be 
divisible  by  the  same  factor.     By  Art.  361, 

Log.  (1-f  :^)-  log.  (1+^)=  log.  (  ^  )  -  log.  (  1+^  ). 


as  a  sinsrle  aua 

+2; 


X — z 
Now,  regarding as  a  single  quantity,  we  may  assume 

1  -\-Z 


/  ^       X—Z  \         .   X—Z  ,   ^/  X—Z  \2   ,   ^  /  X—Z  \3 

Substituting  this  for  log.  (1-|-^) —  log-  (l+2^)j  in  equation  (3), 
and  dividing  both  sides  by  x — z,  we  obtain 

=zA-{-B{x-\-z)-{-C(x2-\-xz-]-z^)-^,  etc. 

Since  this  equation  is  true  for  all  values  of  X  and  z,  it  must  be 
true  when  x=zZ.     Making  this  supposition,  we  have 

A.-— -  =A+2Ba:4-3Crr2-|-4Da:3-f5Ea;4+,  etc.; 

or,  performing  the  division  of  1  by  l+^j  we  have 

A(l— a:+x2_ic3-frt— .  .  .  .  ).z-A+2Ba;-f 3Ca:2+4Da:^+. 
Equating  the  coefficients  of  the  like  powers  of  x  (Art.  314), 
A=A,  B=_^,    c4    D=-^ 

The  law  of  this  series  is  obvious,  the  coefficient  of  the  nth  term 
being  ±— ,  according  as  n  is  odd  or  eve7i. 

A.         A  A 

Hence,  log.  (l-fa:)— Aa;— ^-a:2-f-a:3— -x^-f     .... 

a;2     ic^     x\  X'     x^'  ,  .     ^.^ 

=A(=^-2+T-4  +  5-6-+    •    •    •    )     W 

There  still  remains  one  quantity,  A,  undetermined.  This  is  as  it 
should  be,  for  the  logarithm  of  a  given  number  is  indeterminate 
unless  the  base  of  the  system  be  given. 


336  RAY'S  ALGEBRA,  SECOND  BOOK. 

The  value  of  A  depends  on  the  base  of  the  system,  so  that  when 
A  is  given,  the  base  may  be  determined ;  or,  when  the  base  is 
known,  A  may  be  determined. 

If  we  denote  the  series  in  the  parenthesis  in  equation  (4)  by  rc^, 
we  may  write 

Log,  {l-{-x)=Ax^.     Hence, 

The  logarithm  of  a  number  consists  of  two  factors^  one  of 
rchich  depmids  on  the  number  itself^  and  the  other  on  the 
base  of  the  system  in  which  the  logarithm  is  taken. 

That  factor  which  dej>ends  on  the  base  is  called  the 
Modulus  of  the  system  of  logarithms. 

Lord  Napier,  the  inventor  of  logarithms,  assumed  the 
modulus  equal  to  unity,  and  the  system  resulting  from  such 
a  modulus,  is  called  the  Noperian^  or  Hyperbolic  system. 

For  all  values  of  x  above  x=\.  the  series  (5)  diverges,  and  is,  there 
fore,  inapplicable. 

Designating  the  logarithms  in  this  system  by  log^.,  we  have 

Log^  (l+^)=i-f +  ^-T+'  ''''     (^^ 

Thus,  if  a:^0,  we  find  log^.  1=^0,  as  in  Art.  367. 
If  we  make  x=l^  we  have 


Log^  2^1-1+1-1  +  1-, 


etc. 


374.  The  preceding  series  converges  so  slowly  that  it 
would  be  necessary  to  take  a  great  number  of  terms  to 
obtain  a  near  approximation.  But  we  may  obtain  a  more 
converging  series  in  the  following  manner : 

Resuming  equation  (5), 

Log'.  (1+^)=^-^+^-^  +  ^-,  etc.  .  .  .  (5). 

Substituting  — x  for  a:,  in  this  equation,  we  obtain 

Log'.  (l_^)=_f-f -f-^_f -,cto.  .  (6). 


LOGARITHMS.  S37 

Subtracting  equation  (6)  from  (5),  and  observing  that 

Log^  (l+iC) —  log^.  {l—x)=z  log^l  ■—-   \,  we  have 

,    1+x        I  X     x^      x'^      x^      x\  \ 

lA-x    ^  ,    2x      ,      14-x    ^  ,  1  1 

Since  _n^l4---,    let  -1^=1+-,    .-.   x. 


l^x       '  1—x'         1—x       '  0'   '  ■         20+1' 
and  log^.  ^^  log^  {^^\)=  ^<-  (  ^  ) 

=  log^.    {Z^l)—  log^   2;. 

By  substitution,  the  preceding  series  becomes 
Log-.  (.+l)-log',.=2{2^  +  3^3  +  g^,+  ..    }, 

Log'.  (.+1)=  log'.  z^2{^^  +  p^,3  +  5(2iW+  •  •  ^'^• 

375.  By  means  of  this  series,  the  Naperian  logarithm 
of  any  number  may  be  computed,  when  the  logarithm  of 
the  preceding  number  is  known.  But  the  log',  of  1  is  0, 
(Art.  36Y) ;  therefore,  making  z=l,  2,  4,  6,  etc.,  we  ob- 
tain the  following 

NAPERIAN,  OR  HYPERBOLIC  LOGARITHMS. 
Log^  2=.log^.  1+2  {1  +  ^,  +  ^,  +  ^'^^+   .  . 1^0.693147 

Log^  3=.log^  2+2(^  +  3^^+ J^,  + J^,-|-  .  .}=1.098612 

Log^  4=2.  log.  2 =1.386294 

Log^  5=log^  4+2{^  +  3ip  +  ^.  + J^,+  .  .}=1.609438 
Log^.  6=log^  2+  log^.  3 =1.791759 

Log^7=log^.  6+2  {^^  +  3-433+5^13^+     .     .      }=1.945910 
2d  Bk.        29* 


338  RAYS  ALGEBRA,  SECOND  BOOK. 

Log^.     8.^3  log^.  2,  or  log^  2+  log^  4  .     .     .     .     =.-2.079442 

log^.     9=2  log'.  3 =2.197225 

Log'.  10^  log^  2+  log^.  5 =2.302585 

In  this  manner  the  Naperian  logarithms  of  all  numbers  may  be 
computed. 

When  the  numbers  are  large,  their  logarithms  are  computed  more 
easily  than  in  the  case  of  small  numbers.  Thus,  in  calculating  the 
logarithm  of  101,  the  first  term  of  the  series  gives  the  result  true  to 
seven  places  of  decimals. 

3T6.  To  explain  the  method  of  computing  common  loga- 
rithms from  Naperian  logarithms. 

We  have  already  found  (Art.  373,  Equation  4), 

.,    ,      ,        ,   I  X       X^       X^       X^       X-'       x^  \ 


Denoting  the  Naperian  logarithm  by  an  accent,  -yve  have 
X      x^      x^      X*      x'"      x^ 

+  5-T+-    ■    • 


Log^.  (l+a:)=A^^j--  +  -- 


Since  the  series  in  the  second  members  are  the  same,  we  have 
Log.  (1+a:)  :  log^  (1+^^)  :  :  A  :  A'.     Therefore, 

Tlie  logarithms  of  the  same  number,  in  two  different  si/s- 
ie77is,  are  to  each  other  as  the  moduli  of  those  systems. 

But  in  Napier's  system  the  modulus  A'--l.     Therefore, 
Log.  (l-l-:r)=A  log'.  (1+a;).     Hence, 

To  find  the  common  logarithm  of  any  numher,  multiply  the 
Naperian  logarithm  of  the  number  by  the  modulus  of  the 
common  system. 

It  now  remains  to  find  the  modulus  of  the  common  system. 
From  the  equation,  log.  (l-\-x)=A.  log',  (l+ic). 

Wo  find A=!2?i41+£)      „e„„, 

log'-  (1  f «) 


LOGARITHMS.  339 

The  modulus  of  the  common  system  is  equal  to  the  common 
log.  of  any  number  divided  hy  the  Naperian  log.  of  the  same 
Humher. 

But  the  common  logarithm  of  10  is  1,  and  we  have  calculated  the 
Naperian  logarithm  of  10,  (Art.  375);  therefore, 

log.  10  _        1       _  4342944 

^"l<riO  -  2:302585—^^^^^^' 

which  is  the  modulus  of  the  common  system. 

Hence,  if  N  is  any  number,  wc  have 

Com.  log.  N=r.4342944x  Nap.  log.  N. 

On  account  of  the  importance  of  the  number  A,  its  value  has  been 
calculated  with  great  exactness.     It  is 

A=r.434294481 90325182765. 

3TT-  To  calcxdate  the  common  logarithms  of  numbers 
directly. 

Having  found  the  modulus  of  the  common  system,  if  we  multiply 
both  members  of  equation  (7),  Art.  374,  by  A,  and  recollect  that 
AX  Nap.  log.  N=  com.  log.  N,  the  series  becomes 

Log.(.+i)=iog..+2A{2^^  +  3^j-3  +  g^2iq:iy+  •  ■  ^ 

Or,  by  changing  Z  into  P,  for  the  sake  of  distinction,  and  putting 
B,  C,  D,  etc.,  to  represent  the  terms  immediately  preceding  those 
in  which  they  are  used,  we  have 

Log.(F+l)=log.P4-_lA_+,        ^  3C 


2P+1  ^3(2P+1)2^5(2P+1)2 

,        5D  7E  9j; 

"^7(2P+l)2"^9(2P+l)2"^ll(2P-fl)2'^' 

We  shall  now  exemplify  its  use  in  finding  the  logarithm  of  2. 
Here,  P=:l,  and  2P+l=r3. 


340  RAY'S  ALGEBRA,  SECOND  BOOK. 

Log.  P  =  log.  1 =.00000000; 

_|A_  ^.86858896 ^^^^^^^^,^     ^^^ 

sl^W    -%W  ••••••   — .    (C.) 

5-^1,     -^-^^-^ =,000n480MI>.) 

5D  5X.00071489  nnnn...-.  ,^x 

TiWTT?     =^XS^ =.000056.4;     (E.) 

7E  _7X.00005674  -00000400.     (F  ^ 

pF+Tp 9xP -.00000490,     (F.) 

9F  9X.00000490  fv^nf^nf,iK      /n  ^ 

iT(2P+I7  =^11X3^ =.00000045;     (G.) 

IIG  11X-0QQ00045  00000004.    rn^ 

Therefore,  common  logarithm  of  2      .     .     =^.30102999. 

Exercise. — In  a  similar  manner  let  the  pupil  calculate 
the  common  logarithms  of  3,  5,  V,  and  11. 

For  the  results  to  6  places  of  decimals,  see  the  Table,  page  326. 

3TS*  To  Jind  the  base  of  the  Naperian  system  of  loga- 
rithms. 

If  we  designate  the  base  by  e,  we  have,  (Art.  376), 
Log.  e  :  log',  e  :  :  A  :  A'. 

But  A=r=.4342944,  A'=l,  and  log'.  c=\,  (Art.  367);  hence, 
Log.  c  :  1  :  :  .4342944  :  1 ;  whence,  log.  ^^=.4342944. 

Taking  the  number  of  which  the  logarithm  is  .4342944,  from  the 
table  of  common  logarithms,  we  find  e=2.71828182. 

TVe  thus  see  that  in  both  the  common  and  the  Naperian 
systems  of  logarithms,  the  base  is  greater  than  unity. 

Napier's  logarithms  are  used  in  the  Calculus,  but  not  in 
the  common  operations  of  multiplication,  division,  etc. 


POSITION.  341 

3TO.  The  student  may  prove  tlie  following  theorems  : 

1.  No  system  of  logarithms  can  have  a  negative  base,  or 
have  unity  for  its  base. 

2.  The  logarithms  of  the  same  numbers  in  two  different 
systems  have  the  same  ratio  to  each  other. 

3.  The  difference  of  the  logarithms  of  two  consecutive 
numbers  is  less  as  the  numbers  themselves  are  greater. 

SINGLE    AND    DOUBLE    POSITION. 

Note. — This  subject  is  introduced  in  connection  with  that  of 
logarithms,  because  the  rule  for  Double  Position  is  applied  to  the 
solution  of  exponential  equations. 

380.  Single  Position.— The  Rule  of  Single  Position 
is  applied  to  the  solution  of  questions  which  give  rise  to 
an  equation  of  the  form 

ax=m         (1). 

If  we  assume  x/  to  be  the  value  of  x,  and  denote  by  m' 
the  result  of  the  substitution  of  x'  for  x,  we  have 

ax'=m'         (2). 
Comparing  equations  (1)  and  (2),  we  have 
m' :  m  :  :  ax' :  ax  :  x' :  x ;  that  is, 

As  the  result  of  the  supposition  is  to  the  result  in  the  ques- 
tion, so  is  the  supposed  number  to  the  number  required. 

Example. — What  the  number,  whose  third,  fourth,  and 
sixth  part  being  added,  the  sum  will  be  45  ?     Ans.  60. 

381.  Double  Position. — In  Double  Position,  the  result, 
although  it  is  dependent  on  the  unknown  quantity,  does 
not  increase  or  diminish  in  the  same  ratio  with  it. 


342  RAY'S  ALGEBRA,  SECOND  BOOK. 

The  class  of  questions  to  which  it  is  particularly  appli- 
cable, gives  rise  to  an  equation  of  the  form 

ax-\-h=m         (1). 

If  we  suppose  x'  and  x"  to  be  near  values  of  X,  and  e'  and  e"  to 
be  the  errors,  or  the  differences  between  the  true  result  and  the  re- 
sults obtained  by  substituting  x'  and  a;''  for  x,  we  have 

ax' -^-b^m^e'        (2), 
ax"-\-b=7n-\-e"        (3). 

If  we  subtract  equation  (1)  from  (2),  and  (3)  from  (2),  we  have 

a{x'—x  )=e'  (4), 

a{x'—x")=e'—e''    (5). 


From  these  equations,  we  easily  obtain 


(6). 


e'—e"        e' 
By  subtracting  equation  (1)  from  (3),  we  also  find 

a{x"—x)z=e'\  and  thence, 
x'—x"     x"-~x 

Hence,  (Art.  263),  Tlie  difference  of  the  errors  is  to  the 
difference  of  the  two  assumed  numbers,  as  the  error  of  either 
result  is  to  the  difference  between  the  true  result  and  the  cor- 
responding assumed  number. 

When  the  question  gives  rise  to  an  equation  of  the  form  ax-\-b^=z7n, 
this  rule  gives  a  result  absolutely  correct;  but  when  the  equation 
is  of  a  less  simple  form,  as  in  exponential  equations  (Art.  383),  the 
result  obtained  is  only  approximately  true. 

Corollary. —The  common  arithmetical  rule  is  deduced 
from  the  following  value  of  x,  found  either  from  equa- 
tion (6)  or  (7) : 

_e'x"—e"x' 


EXPONENTIAL  EQUATIONS.  343 


EXPONENTIAL     EQUATIONS. 

382.  An  Exponential  Equation  is  an  equation  in 
which  the  unknown  quantity  appears  in  the  form  of  an 
exponent  or  index ;  as, 

a-^=6,  x*=:a,  a^^=^c^  etc. 

Such  equations  are  most  easily  solved  by  means  of  loga- 
rithms. 

Thus,  in  the  equation     .     .     .     a^=6, 

We  have  (Art.  362),    .     .     x  log.  a=  log.  6; 

^  log.  b 

Or, 2;=-^ — . 

log.  a 

Ex.  1. — What  is  the  value  of  x  in  the  equation  2'=64? 
Here, x  log.  2=  log.  64 ; 

log.  64      1.806180     ^     ^ 
Whence,     .     .     .     x^^^  = -^^^^=Q,  Ans. 

383.  If  the  equation  is  of  the  form  a;^=a,  the  value 
of  X  may  be  found  by  Double  Position,  as  follows : 

Find  by  trial  two  numbers  nearly  equal  to  the  value 
of  X ;  substitute  them  for  x  in  the  given  equation,  and 
note  the  results.  Then,  from  (7),  we  have  (Art.  263)  the 
proportion, 

As  the  difference  of  the  errors,  is'^  the  difference  of  the  two 
assumed  numbers,  so  is  the  error  of  either  result,  to  the  correc- 
tion to  he  apjylied  to  the  corresponding  assumed  number. 

Ex.  1.— Given  x'^^lOO,  to  find  the  value  of  a;. 

The  value  of  x  is  evidently  between  3  and  4,  since  3^=27,  and 
4'=256;  hence,  taking  the  logarithms  of  both  sides, 

X  log.  x=  log.  100:=2. 


344  RAY'S  ALGEBRA,  SECOND  BOOK. 

By  trial,  we  readily  find  that  x  is  greater  than  3.5,  and  less  than 
3.6:  then,  let  us  assume  3.5  and  3.6  for  the  two  numbers. 


First  Supposition. 

x=3.3-         log.  a:=.544068 
multiply  by  3.5,  we  find 
X  log.  X  =1.904238 

true  no.  =2.000000 


=—.095762 


Second  Supposition. 

x=S.6;         log.  a;=.556303 
multiply  by  3.6,  we  find 
X.  log.  X  =2.002690 

true  no.  =2.000000 


error  -f  .002690 


DiflF.  results  :  DifF.  assumed  nos.  :  :  Error  2d  result :  Its  cor. 
.098452      :  0.1  :  :         .002690        :  .00273 

Hence,        3.6 -.00273=3.59727  nearly. 

By  trial  we  find  that  3.5972  is  less,  and  3.5973  greater  than  the 
true  value;  and  by  repeating  the  operation  with  these  numbers,  we 
would  find  a;=3.5972849  nearly. 

2.  Given  20"=:100,  to  find  x.         Ans.  a:r:=1.53724. 

3.  Given  x^=b,  to  find  x.  Ans.  a:=2.129372. 

4.  How  many  places  of  figures  will  there  be  in  the  num- 
ber expressing  the  64'''  power  of  2  ?  Ans.  20. 

5.  Given  a^-'+'^=c,  to  6nd  x.   Ans.  ^_  ^g-  ^~  .  •    ^g-  ^ 

6.  log.  a 

6.  Given  a'^^Z/"'=c,  to  find  x. 

Ans.  x=  — "j r. 

m.  log.  a-j-  *i-  log.  o 

7.  Given  x-\-y=a,  and  in^y=n,  to  find  x  and  y. 

Ans.  x=^(a-\-  log.  ?ih-  log.  m),  y=^l{<oi —  log.  n-^-  log.  m). 

8.  Given  2^3^^2000,  and  ^z=bx,  to  find  the  values 

of  X  and  z. 

Ans  ^__3(3+Jog^i)_  5(3+  log.  2) 

ADS.  a;_g  2       -  g,  s_g  21^^  j^^^    g. 


INTEREST  AND  ANNUITIES.  345 

2  loiT.  2 


9.  Given  a?^ — 2a-^=8,  to  find  x.       Ans. 


loo;,  a 


Suggestion, — This  is  a  quadratic  form,  therefore  complete  the 
square. 

10.  Given  22^4-2^=^12,  to  find  x.    Ans.  a;==:l. 58496. 

11.  Given  a^-\ — =6,  to  find  x. 


log.  a. 

12.  Given  xy^^y''^  and  x?=^y\  to  find  a^  and  y. 

Ans.  a;=2{,  5^=r=3|. 

13.  Given  (a2_^,2>)2fx-i)^(^_j>)2x^  to  find  x. 

Ans.  .^1+l^M. 
^log.  (a-f  6) 

14.  Given  (a*— 2a262-f£*)'"'=(«— ^)'^(«+^)~'5  *«  ^^"^  ^^ 

Ans   ^^^Qg-  (^-^) 
log.  (a4-6)* 

15.  Given  xy^=y^^  and  x^=^yi^  to  find  a?  and  y. 


Ans...=  (|jA,    j=(|| 


P-9. 


IG.  Given   3^^'     ^"^+^^=1200,  to  find  x. 


Ans.  a::r-4.33,  or  —0.33. 


INTEREST    AND    ANNUITIES. 

3^4.  The  solution  of  all  questions  in  Interest  and 
Annuities  may  be  simplified,  and  also  generalized,  by 
means  of  algebraical  formulae ;  but  certain  problems  in 
Compound  Interest  and  Annuities  may  be  very  much 
abridged  by  the  use  of  logarithms. 


346  RAYS  ALGEBRA,  SECOND  BOOK. 

Let  P=  the  principal,  or  sum  at  interest  in  dollars. 
r=  the  interest  of  1  ^  for  one  year. 
t=  the  time  in  years  that  P  draws  interest. 
A=  the  amount,  at  the  end  of  t  years. 

385.  Simple  Interest. — Since  tr  represents  the  inter- 
st   of   1  $   for  t   years,  and   P^r,  the  interest  of  P  $  for 

I  years ;  therefore, 

A==P-j-P/r=P(l  +  ^r) (1). 

From  this  equation,  any  three  of  the  quantities  P,  r,  <,  A,  being 
given,  the  fourth  may  be  found.     Thus, 

_A_      ._A— P        _A— P 

Examples  may  be  given  from  any  treatise  of  arithmetic. 

386.  Compound  Interest. — Let  R=l-f  ?-,  the  amount 
of  1  $  for  one  year  ;  then,  R  will  be  the  principal  for  the 
second  year  ;  and  since  the  amount,  in  each  case,  is  pro- 
portional to  the  principal  for  the  same  time ;  therefore, 

1  :  E  :  :  R  :  the  amount  of  1  J  in  2  years:=R2. 
1  :  R  :  :  R^  :  R^,  the  amount  of  1  f  in  3  years. 

And,  in  like  manner,  R'  is  the  amount  of  1  $  in  ^  years. 

The  amount  of  Pf  will  be  P  times  the  amount  of  1  $. 

Hence, 

A=P.R'=P(l-f-r)';  whence, 

Log.  A=  log.  P+^.  log.  (1-f  r)         (1). 

Log.  P=  log.  A-^.  log.  (1+r)         (2). 

log.  A—  log.  P  o 

'-     log.(l+0~  ^^^' 

_.         ,-,   ,    X     log.  A —  log  P  ..^ 

Log.  (l+r)=-^^ ^ ^1-  (4). 

Corollary  L— The  interest  =A— P=PR'— P^P(R'-  1). 


INTEREST  AND  ANNUITIES.  347 

Corollary  2. — If  t^e  interest  is  paid  half-yearly^  tz=2t. 
and  '■=^.     Hence, 

A=p/  1+^  Y{h).   li^2i\di  quarterly,  A=:Vi  1-j-^  ^(6). 

Corollary  3.— From  the  equation  A=rP.R<,  we  can 
readily  find  the  time  in  which  any  sum,  at  compound  in- 
terest, will  amount  to  hvice^  thrice^  or  m  times  itself. 

Thus,  if  A.=2P;     then,  2P=PR<    .-.    R<=2,  <and  t^^^^l^. 

log.  R 

"       if  A=3P;     then,  R<  =3,    and      t=  log.  3  h-  log.  R; 

"      if  Ai=mP;  then,  R<  =m,  and      t=  log.  m-j-  log.  R. 

1.  Let  it  be  required  to  find  the  time  in  which  any  sum 
will  double  itself  at  10  per  cent,  compound  interest. 

Here,  r=.10,  R=l4r=l+.10=:1.10;  hence, 
,     log.  2       .301030    ^„^,, 

^^ioi:R=:o-4i393='-2'^  ^^^•'  ^"^- 

2.  What  is  the  amount  of  1  $  for  100  years  at  6  per 
cent,  per  annum,  compound  interest?        Ans.  $339.28. 

3.  How  many  figures  will  express  the  amount  of  $1  for 
1000  yr.,  at  6  %  per  annum,  comp.  int.?  Ans.  26. 

4.  In  how  many  yr.  will  any  sum  double  itself  at  com- 
pound interest,  at  5,  6,  7,  and  8  %  per  annum  respectively? 

Ans.  14.2066,  11.8956,  10.2447,  9.0064  yrs. 

5.  In  what  time,  at  compound  interest,  reckoning  5  %  per 
annum,  will  $10  amount  to  $100?  Ans.  47.19  yrs. 

6.  If  $P,  at  compound  interest,  amount  to  $M  in  t  years, 
what  sum  will  amount  to  $P  at  the  end  of  t  years? 

387.  The  increase  of  the  population  of  a  country  may 
be  computed  on  the  same  principles  as  compound  interest. 


348  RAY'S  ALGEBRA,  SECOND  BOOK. 

1.  The  population  of  the  United  States  in  1^790  was 
3900000,  and  in  1840,  iVOOOOOO.  Required  the  aver- 
age rate  of  increase  for  each  10  years. 

Here,  there  are  5  periods  of  10  years  each.  Hence,  by  comparing 
the  quantities  given,  with  those  in  equation  (4),  Art.  386,  we  have 

A=17000000,  P=3900000,  and  t=5. 

Log.  A,  (see  table,  page  326), 7.230449 

Log.  P 6.591065 

Divide  by  5 5)0.639384 

Log.  (1+r)  1.342 0.127877 

Hence,  r=l. 342— 1  =.342=34^  per  cent.,  Ans. 

2.  The  population  of  England  in  1820  was  11000000, 
and  in  1830  about  13000000.  What  was  the  annual 
rate  of  increase,  and  in  what  time  would  the  population 
be  doubled?  Ans.  .016  per  cent.,  and   41.49  yrs. 

388.  Compound  Discount. — The  present  value  of  a 
sum  P,  due  t  years  hence,  reckoning  compound  interest,  is 
easily  obtained  from  Art.  386. 

Let  ?'=  the  present  worth,  then  in  t  years,  P'  at  compound  inter- 
est, will  amount  to  P;  therefore, 

P=:P^(l+r)^  .-.  P^-^|~,        (1). 

p 

Let  D=  Comp.  Discount;   then,  D^-P— P^=P ^.     (2) 

From  equation  (1),  log.  P'=  log.  P— ^  log.  (1+r)         (3). 

Ex. — What  is  the  compound  discount  on  $1000,  due 
in  20  years,  at  5  per  cent.?  Ans.  $623.11. 

389.  Annuities  Certain. — An  Annmfi/  is  a  sum  of 
Tnoney  which  is  payable  at  equal  intervals  of  time. 

An  annuity  already  commenced  is  said  to  be  in  posses- 
sion ;  one  commencing  after  a  certain  number  of  years  has 


INTEREST  AND  ANNUITIES.  849 

elapsed,  is  called  a  deferred  annuity,  or  an  annuity  in  re- 
version. 

An  annuity  certain  is  one  limited  to  a  certain  number 
of  years.  A  life  annuity  is  one  which  terminates  with  the 
life  of  any  person.  A  perpetuity^  or  perpetual  annuity,  is 
one  which  is  unlimited  in  its  duration. 

All  the  computations  relating  to  annuities  are  made 
according  to  compound  interest. 

300.  To  find  the  amount  of  an  annuity  in  any  number 
of  years,  at  compound  interest. 

Let  a  denote  the  annuity,  p  the  present  value,  m  the 
amount ;  and  r,  R,  t,  the  same  as  in  the  preceding  articles. 

The  first  annuity  a,  becomes  due  at  the  end  of  the  year,  and  thus, 
in  t — 1  years,  will  amount  to  CfR'— i  (Art.  386).  The  second  and 
third  annuities,  due  in  2  and  3  years,  amount  to  aR'—^  and  aR'— 2, 
and  so  on  to  the  last,  which  is  a. 

Hence,  the  entire  amount  is  the  sum  of  a  geometrical  series, 
whose  first  term  =:aR'— ^,  common  ratio  =R,  and  last  term  =^a; 
therefore,  by  reversing  the  order  of  the  terms,  we  have 

w=:a+aR+aR2-f  aR3-|- .    .    .    .    -faR<-2-|-aR<-i. 

...(Art.297),m=«5^  =  „ll±!:)!=^. 
^  ^'  R— 1  r 

If  the  annuity  is  to  be  received  in  half-yearly  installments, 

„.    ,  a(l+Ar)2<_l         (l-j-ir)2<-l 

We  have         m^^.     ^V- =«• . 

2  -hr  r 

If  quarterly,    m=j.l— 1^— ^a.'—^^—^ . 


Corollary. — Similarly,  the  amount  of  a  dollars  placed 
out  annually  for  t  successive  years,  at  compound  interest, 
would  be 

^( 1 


350  RAY'S  ALGEBRA,  SECOND  BOOK. 

1.  To  what  sum  will  an  annuity  of  $120  for  20  years 
amount  at  6  per  cent,  per  annum?  Ans.  $4414.27. 

2.  Three  children,  A,  B,  C,  who  come  of  age  at  the  end 
of  a,  6,  c,  years,  are  to  have  $P  divided  among  them,  so 
that  their  shares  being  placed  at  compound  interest,  each 
shall  receive,  at  coming  of  age,  the  same  sum.  Find  the 
share  of  A,  the  youngest.  P 

l4-Il«-^-fR«--- 

3.  What  would  $100,  put  out  annually  at  compound 
interest,  amount  to  in  10  years  at  6  per  cent.? 

Ans.  $139Y.1G. 

391.  To  find  the  present  value  of  an  annuity  to  he 
paid  t  years^  at  compound  interest. 

Let  p  denote  the  present  value  of  the  annuity  a\  then,  the 
amount  of  p$  in  t  years  =:pR<  (Art.  386),  and  the  amount  of  the 

R'— 1 

annuity  a  in  the  same  time  is  (Art.  390)  a.- — j  ;  but  these  two 

amounts  must  be  equal  to  each  other;  hence,  wo  get 

R<— 1        ^  R<— 1  a     /.      1   \ 

^^-«-r:=i'  ^"^  ^-«Rp:=r)=Ri3  ( 1-R'  )• 

Corollary. — If  the  annuity  is  to  continue  forever,  t  be- 
comes   infinite,  =r-,  vanishes,  and  we  have  p=^ — t=~- 

1.  What  is  the  present  worth  of  an  annuity  of  $250, 
payable  yearly  for  30  yr.  at  5  %  ?       Ans.  $3843.1135. 

2.  What  is  the  present  worth  of  a  perpetual  annuity  of 
$600  at  6  %  per  annum?  Ans.  $10000. 


392.  To  find  the  present  value  of  an  annuity  in  rever- 
sion; that  is,  an  annuity  which  is  to  commence  at  the  end 
of  n  years,  and  to  continue  t  years. 


GENERAL  THEORY  OF  EQUATIONS.  351 

Bj  Art.  391,  the  present  value  of  the  annuity  for  7i-\-t  years,  is 

The  difference  of  these  two  sums  is  the  value  in  reversion ; 
therefore, 

■^~lt— 1  \  ir      K»+«  )~rlV'  \        Rt  J' 

If  the  annuity  is  payable  forever  after  n  years,  we  have 

^~rR^' 

1.  What  is  the  present  value  of  an  annuity  of  $112.50, 
to  commence  at  the  end  of  10  years,  and  to  continue  20 
years,  at  4  %  ?  Ans.  $1032.877. 

2.  What  is  the  present  value  of  an  annuity  of  $1000, 
to  commence  at  the  end  of  15  years  and  continue  forever, 
at  6  %  per  annum?  Ans.  $6954.40. 

3.  A  deht  of  a$,  accumulating  at  compound  interest,  is 

discharged  in  ?i  years,   by  equal  annual  payments  of  6$; 

find  the  value  of  n.  .  log.  h —    log.  (b — ra) 

Ans.  n=z  , i^  ,  \ -. 

log.  (l+r) 

4.  A  debt  of  $8000,  at  6  %  compound  interest,  is  dis- 
charged by  eight  equal  annual  payments.  Required  the 
annual  payment.  Ans.  $1288.286. 


XII.  GENERAL  THEORY  OF 
EQUATIONS. 

393.  From  Art.  113,  it  is  obvious  that,  as 
ax-\-b=zO,  is  an  equation  of  the  1st  degree, 
x'^-\-hx-\-c=0,  is  an  equation  of  the  2d  degree, 
x^-\-bx^-^cx-\-d=Q,  is  an  equation  of  the  3d  degree;  so, 

a;«+Ax«-^-(-Ba;"-2-f-Ca;"-'-f ^Tx-\-Y=0,  is,  in 

general,  an  equation  of  the  n'^  degree. 


352  RAY  S  ALGEBRA,  SECOND  BOOK. 

The  coefficients,  A,  B,  C,  etc.,  may  be  positive  or  nega- 
tive, integral  or  fractional ;  and  any  of  them  may  be  equal 
to  zero. 

If  the  coefficient  of  the  highest  power  of  x  is  not  unity, 
it  may  be  made  so  by  division. 

394.  A  Root  of  an  equation  is  a  number,  or  quantity, 
Buch  that  being  substituted  for  the  unknown  quantity,  the 
equation  will  be  verified. 

Thus,  3  is  a  root  in  the  equation  x^-\-2x^ — 14ic — 3=0. 

A  Function  of  a  quantity  is  any  expression  dependent 
on  that  quantity.     Thus,  2x-{S  is  a  function  of  x; 
t)x^,  is  a  function  of  a: ; 
7x — 3y^,  is  a  function  of  x  and  y. 

In  a  series,  when  the  signs  of  two  successive  terms  are 
alike,  they  constitute  a  permanence,  when  they  are  unlike, 
a  variation. 

Thus,  in  the  polynomial,  —r — s-^t-\-u,  the  signs  of  the  first  and 
second  terms  constitute  a  permanence,  of  the  second  and  third  a 
variation,  and  of  the  third  and  fourth  a  permanence. 

395.  Proposition  I. — If  a  is  a  root  of  any  equation^ 

cc"+Ax"-i-f  Bx"-2+Ca:"-3+.    .  .  .  +Ta;+V=^0,  («.), 

then  will  the  equation  he  divisible  hy  x — a. 

For  if  a  is  one  value  of  a:,  the  equation  will  be  verified  when  a  is 
substituted  for  X.     This  gives 

a"+Aa"-i-fBa"-2-|-Ca"-3-i-.    .    .    .    -|-Ta+V:=0; 
Or,  V=— a'*— Aa"-i— Ba"-2_Ca"-3—    .    .    .    — Ta. 

Substituting  this  value  of  V  in  the  given  equation,  and  arranging 
the  terms  accoi'ding  to  the  same  powers  of  x  and  a,  we  have 

(x''_a")+A(a;'»-i— a''-i)+B(a:"-2— a'^-z)-!-.  _  ^  ,    _|.T(a:— a)=0. 

As  (Art.  83)  each  of  the  expressions  (x** — a"),  (a:"-! — «""')>  ^tc, 
is  divisible  by  X — a,  the  given  equation  is  divisible  by  x — a. 


GENERAL  THEORY  OF  EQUATIONS.       353 

Corollary. — Conversely,  if  the  equation 

^u_|_^^H-i_|_B^n-2_j_^  .  .  .  -^T:r-f  V=:=0,  (n)  is  divisible 
by  X — a,  then  a  is  a  root  of  the  equation. 

For  if  the  equation  (n)  is  divisible  by  X — a,  if  we  call  the  quo- 
tient Q,  we  have  (x — a)Q=0  (n),  v/hich  may  be  satisfied  by  mak- 
ing X — a=0,  whence  x=:a. 

D'Alembert's  Proof  of  Prop.  I, — If  said  division  leave  a  re- 
mainder, call  it  R,  and  the  quotient  Q;   then,  equation  [n)  becomes 

(a:— a)Q4-R=0. 

But  X — Cf=0,  .-.  R=zO;  that  is,  there  is  no  remainder  on  dividing 
equation  (n)  by  x — a. 

Illustration. — In  the  equation  x^-j-x"^ — 14x — 24=:0, 
the  roots  are  — 2,  — 3,  and  4  ;  and  the  equation  is  divis- 
ible by  x-\-2,  x-\-S,  and  x — 4. 

396.  Proposition  II.-— ^71  equation  of  the  n'^'  degree 
has  n  roofs. 

Let  a  be  a  root  of  the  equation 

cc'^-f  Aa;'»-i4-Ba:"-2-f-Ca:"-'5+.    .     .    .     ^-Tic-f  V=0  (n). 

By  Art.  395  this  equation  is  divisible  by  X — a.  If  we  perform 
the  division,  and  denote  by  Aj,  Bj,  etc.,  the  coefficients  of  the 
powers  of  X  in  the  quotient,  equation  (n)  becomes 

(a:— (2)(a;"-i+Aia:«-24-Bia;"-34-,    .    .    .    -|_Tia;+Vi)=^0. 
Hence,  x"-^-\-A^x''~^-^B^x''-^^ ^Tia:-f  Vi=0. 

Now,  this  equation  must  also  have  a  root,  which  may  be  denoted 
by  6,  and  is  (Art.  395)  divisible  by  x — b.     Hence, 

(a:— 6}(u;""2^A2a;*»-3+B2a;"-4+.    .    .    .     +t2^+V2)=0. 

Placing  the  second  factor  of  this  equation  equal  to  zero,  taking  c, 
a  third  root,  and  dividing  by  X — C,  we  shall  have  an  equation  of  a 
degree  still  lower  by  a  unit. 

It  is  evident  that  if  this  operation  be  continued,  the  exponent  n 
will  be  exhausted,  and  the  last  quotient  will  be  unity;  hence,  call- 
ing the  last  root  I,  we  shall  have 
2d  Bk.  30 


354  RAY'S  ALGEBRA,  SECOND  BOOK. 

{x—a){x—b){x—c){x—d),  .  .  .  {x—l)=0,  which  is  satisfied 
by  making  a:=a,  b,  c,  d,  .  .  .  .  or  Z;  that  is,  the  equation 
has  n  roots,  a,  6,  c,  d,  etc. 

Corollary  I. — If  we  know  one  root  of  an  equation,  by 
dividing  (Art.  395)  we  may  find  the  equation  containing 
the  remaining  roots. 

Thus,  one  root  of  the  equation  a:^— 12a;24-47a:— 60=0,  is  5,  and 
by  dividing  it  by  x—6,  the  quotient  is  a:^— 7a;-f  12=0,  the  roots  of 
which  may  be  found,  viz.,  -)-3  and  -|-4. 

Corollary  II. — When  any  equation,  whose  right  hand 
member  is  zero,  can  be  separated  into  factors,  the  roots 
of  the  equation  may  be  found  by  placing  each  of  the  fac- 
tors equal  to  zero. 

Thus,  if  a:2-f4a:=0,  we  have  a:(a:-|-4)=0,  whence  a:=0,  and 
x=—4.     (See  Art.  253.) 

1.  One  root  of  the  equation  x^ — llrr'^-l-^Sic-j-SS^^O  is 
— 1  ;  find  the  equation  containing  the  remaining  roots. 

Ans.  3^2— 12.T-f  35:^.0. 

2.  One  root  of  the  equation  x^ — 9x'^-\-26x — 24=::0  is 
3  ;  find  the  remaining  roots.  Ans.  2  and  4. 

3.  Two  roots  of  the  equation  x'-j-2x^—41x''—42x-\-SQ0 
=-0,  are  3  and  — 4 ;  required  the  remaining  roots. 

Ans.  5  and  — 6. 

Remark. — Two  or  more  of  the  n  roots  may  be  equal  to  each 
other.     Thus,    the   equation   x^ — 6x^-\-l2x — 8=0,    is   the   same   as 

(a;_2)(a:— 2)(a;— 2)=0,  or  {x — 2)3=0.     Hence,  the  three  roots  are 
x=2,  x=2,  and<<r=2. 

397.  Proposition  III. — A"o  equation  of  the  n^^  decree 
can  have  more  than  n  roots. 

If  it  be  possible  let  the  equation 

a:»»+Ax«-^+Bx»-2-f  Ca;"-3-}-.  .  .  .  -j-Ta;-}-V=0, 


GENERAL  THEORY  OF  EQUATIONS. 


355 


besides  the  n  roots,  a,  i,  r,  c?,  etc.,  have  another  root,  r,  not  identical 
with  either  of  the  roots  a,  6,  c,  c?,  etc.;  then,  the  equation  must  be 
divisible  by  x — r  (Art.  395) ;  this  gives 

a;"+Aa:"-i+Ba:"-2+,  etc.,  =(a;— r)(x"-i+A^:r"-2+,  etc.,)  or 

{x—a){x—b){x—c).  .  .  (a;-0^(^— ^)(^"~^+^^^"~^+'  ^^^O 

But  since  r  is  a  value  of  X,  we  have,  by  substitution, 

(r—a){r-b){r—c).  .  .  {r-l)  =(r— r)(a:'^-i+A^a:"-2+,  etc.) 

Now,  the  second  member  of  this  equation  is  ==iO,  because 
^r — r):=0;  but  the  other  side  can  not  be  0,  since  r  is  not  equal  to 
any  of  the  quantities  a,  6,  C.  etc.;  hence,  the  supposition  is  absurd 
that  X  can  have  any  value  other  than  a,  b,  C,  d,  .  .  I. 


30S.  PropositiOTl  J.V. —  To  discover  the  relations  hetw( 
the  coefficients  of  an  equation  and  its  roots. 

Let  Xz=.a,  \  Then, 

x=zb, 


-d,  etc. 


^  Then,       (  x—a=0, 
I  J  x-b=0, 

f  j  x-c=.0, 

)  I  x—d^O,  etc. 


By  multiplying  together  the  corresponding  terms  of  the  last  set 
of  equations,  we  have  {x — a){x—b){x — c){x — d)—0. 

If  we  perform  the  actual  multiplication  of  the  factors,  we  find 


X*- 


a 

x^-i^ab 

x^—abc 

-b 

-\-ac 

—abd 

-c 

^ad 

^acd 

d 

+6c 
^cd 

—bed 

xA^abcd  ^ 


0. 


Similarly,  in  the  equation  of  the  n'^'  degree, 
a:"+Aa;"-i  +  Ba;"-24-,  etc.,  ={x—a){x—b){x- 


(o;— Z)=0. 


If  we  perform  the  multiplication  of  the  n  factors,  we  shall  have 
— a—b—c^  etc.,  =A;  a6+ac-|-ad,  etc.,  =B;  —abc—abd—acd, 
etc.,  ^C;  and  so  on.     For  the  last  term  •±zCLbcd  .  .  .  kl^=Y. 

The  db  is  prefixed  to  the  last,  or  absolute  term,  because  the  prod- 
uct — ay<^—by  —  c.  .  .  .  X — ^)  will  be  plus  or  minus,  according  as 
the  degree  of  the  equation  is  even  or  odd.     Hence, 


350        RAY  S  ALGEBRA,  SECOND  BOOK. 

1.  The  coefficient  of  the  second  term  of  any  equation  is  equal 
to  the  smn  of  all  the  roots,  with  their  signs  changed. 

2.  The  coefficient  of  the  third  term  is  equal  to  the  sum  of 
the  products  of  all  the  roots  taken  two  and  two. 

3.  The  coefficient  of  the  fourth  term  is  equal  to  the  sum  of 
the  products  of  all  the  roots  taken  three  and  three,  with  their 
signs  changed.      And  so  on  ;    and 

4.  The  last  term  is  the  product  of  all  the  roots,  with  the 
sign  changed  if  the  degree  of  the  equation  is  odd. 

Corollary  I. — If  any  term  is  toanting,  its  coefficient  is  0. 

II.  If  the  2'^  term  is  wanting,  the  sum  of  the  roots  is  0. 

III.  If  the  3*^  ter77i  is  wanting,  the  sum  of  the  products  of 
the  roots,  taken  two  and  two  in  a  product,  is  0. 

IV.  If  the  absolute  term  is  wanting,  the  product  of  the  roots 
must  he  0,  and  hence  one  of  the  roots  must  be  0. 

V.  Since  the  last  term  is  the  j^roduct  of  all  the  roots,  there- 
fore, it  must  be  divisible  by  each  of  them;  that  is,  every  ra- 
tional root  of  an  equation  is  a  divisor  of  the  last  term. 

EXAMPLES    ILLUSTRATING    THE    PRECEDING   PRINCIPLES. 

1.  Form  the  equation  whose  roots  are  3,  4,  and  — 5. 

The  equations  X—3,  x=4,  and  a;=— 5,  give  a:— 3z=0,  re— 4=0, 
and  a;4-5::=:0;  hence,  {x—3){x—4){x-{-b)=zX^—2x^—23x-\-60^0. 
Here,  3-|-4 — 5=-|-2,  coefficient  of  2'^  term  with  contrary  sign. 

3x4+3X— 5+4X— 5=— 23,  the  coefficient  of  the  3'^  term. 

3x4X — ^= — 60,  the  sign  of  which  must  be  changed,  for  the  last 
term,  because  the  degree  of  the  equation  is  odd. 

2.  What  is  the  equation  whose  roots  are  2,  3,  and  — 5  ? 
(See  Cor.  2.)  Ans.  cc^— 19x+30=:0. 

3.  Form  the  equation  with  roots  0,  — 1,  2,  and  — 5. 

Ans.  x*-\-4x'—7x''—10x=0. 

4.  Find  the  equation  whose  roots  are  l=t|/2  and  2=t:|/3. 

Ans.  X*— 6x»+8a;2-f-2a;~l=-0. 


GENERAL  THEOliY  OF  EQUATIONS.  ^57 

6.  What  is  the  4'''  term  of  the  equation  whose  roots  are 
-2,  —1,  1,  3,  4?  Ans.  2^x\ 

399.  Proposition  V. — No  equation  having  unity  for  the 
coefficient  of  the  first  term,  and  all  the  other  coefficients  inte- 
gers^ can  have  a  root  equal  to  a  rational  fraction. 

Assume  that  all  the  coefficients  are  integers  in  the  gen- 
eral equation, 

x»-f  A.x«-i4-Ba)«-2-f _f  Ta:-f  V=0. 

If  possible,  let  7-,  a  fraction  in  its  lowest  terms,  be  a  root  of  this 

equation ;  then,  by  substituting  it  for  iC,  reducing  the  terms  to  a 
common  denominator,  transposing,  etc.,  we  shall  have 

^=— Aa"-i— Ba^-^ft- _Ta6"-2— V6"-i. 

b 

But,  by  hypothesis,  a  and  6,  and,  consequently,  a^  and  6,  contain 
no  common  factor;  therefore,  an  irreducible  fraction  is  equal  to  a 
series  of  integers,  which  is  absurd.  Hence,  the  supposition  is  absurd, 
and  the  equation  has  no  fractional  root. 

400.  Proposition  VI. — If  the  signs  of  the  alternate 
terms  of  an  equation  he  changed,  the  signs  of  all  the  roots  will 
be  changed. 

Let  a  be  a  root  of  the  equation 

a:"+Aa:»-i  +  Bx»-2-fCx»-3-[-.  .  .  .  -j-V=0  (1); 
Then,  a"+Aa«-i-|-Ba«-24-Ca"-'+.  .  .  .  +V=^0  (2). 
Changing  the  signs  of  the  alternate  terms  of  equation  (1), 

^n_^^n_l_|_j3^n_2_C5;n_3_j_ ±V=-0  (3). 

Substituting  — a  for  X  in  this  equation,  we  have 

an_Aa"-i_^Ba"-2— Ca"-3 =tV=0        (4). 

Now,  if  n  be  even,  the  2<^,  4"^,  etc.,  terms  will  contain  odd  powers 
of  — a,  which  will  (Art.  193)  render  those  terms  positive.  Hence, 
the  whole  result  will  be  the  same  as  that  produced  by  the  substitu- 
tion of  a  for  X  in  equation  (I). 


358  RAY'S  ALGEBRA,  SECOND  BOOK. 

But  if  n  be  odd^  the  1*',  3«^,  etc.,  terms  will  be  negative,  which  will 
render  all  the  terms  of  (4)  negative.  Changing  the  signs  of  all  the 
terms,  (4)  becomes  the  same  as  (2), 

Hence,  if  a  is  a  root  of  equation  (1),  — a  is  a  root  of  (3),  whether 
n  be  odd  or  even. 

Ex.— The  roots  of  the  equation  a^— Sa^'— 10x-{-24=0, 
are  2,  — 3,  and  4  ;  what  are  the  roots  of  the  equation 
ic3_|.3^2_i0x— 24=0?  Ans.  -2,  3,  and  —4. 

401.  Proposition  VII. —  When  the  coefficients  of  an 
equation  are  real,  if  it  contains  imaginary  roots,  the  number 
of  these  roots  must  he  even. 

If  a+^j/ — 1  be  a  root  of  the  eq.  ic» — A.x"~*-)-Ba;"~^ 
— Ca3"~^-|-,  etc.,  =0;  then,  a — h-^^ — 1  is  also  a  root. 

In  the  equation,  substitute  a-\-by^ —1  for  X,  and  the  result  will 
consist  of  two  parts: 

1*'.  Possible  quantities  which  involve  the  odd  and  even  powers 
of  «,  and  the  even  powers  of  6]/  — 1 ; 

2'^.  Impossible  quantities  which  involve  the  odd  powers  of  b^/ — 1. 

Call  the  sum  of  the  possible  quantities  P,  and  of  the  impossible 
Q|/^;  then,  P+Q|/^^  is  the  whole  result;  hence,  P-f  Q^IIT=0. 

But  the  first  quantity  being  real,  and  the  second  imaginary,  in 
order  to  satisfy  the  equation,  each  of  the  quantities  must  be  0;  this 
gives  P^O,  and  Q^^^^O. 

Again,  let  a—b^/ — 1  be  substituted  for  X,  and  the  !«'  part  of  the 
result  will  be  the  same  as  before,  and  the  2<^  part,  which  arises  from 
the  odd  powers  of  6y/ — 1,  will  differ  from  the  former  imaginary 
part  only  in  its  sign;  therefore,  the  result  will  be  P — Q^/ — I;  but 
since  P^O,  and  Q^ — 1=0,  we  must  have  P — Q^/  — 1=0. 

Hence,  a — 6/ — 1  is  a  root  of  the  equation,  since  its  substitution 
for  X  gives  a  result  equal  to  0. 

Corollary  I. — In  a  similar  manner,  it  may  be  shown 
that  surd  roots  of  the  form  a±]/6,  ±Z>^/ — 1,  or  ±]/^, 
enter  an  equation  by  pairs. 

Corollary  II. — Since  irrational  and  imaginary  roots  al- 
ways occur  in  pairs  where  the  coefficients  are  real,  it  fol- 


GENERAL  THEORY  OF  EQUATIONS.       359 

lows  that  every  equation  of  an  odd  degree  must  have  at 
least  one  real  root. 


Corollary  III. — Corresponding  to  any  pair  of  imaginary 
roots  a±^^ — 1^  we  have  in  the  eq.  the  quadratic  factor, 

[x—^a^hy'—l) } {x— (a— 6i/^=a)  ]=(x—ay  f  6^ ; 

Hence,  every  eq.  of  an  even  order,  with  real  coefficients,  is 
composed  of  real  factors  of  the  second  degree. 

1.  One  root  of  the  equation  x^ — 26;t-|-60=^0  is  — 6; 
required  the  other  roots.  Ans.  3±|/ — 1. 

2.  One  root  of  a:'— Yo^^-flSx— 3=0,  is  2— 1/3  ;  find 
the  other  roots.  Ans.  2-[-i/3  and  3. 

3.  One  root  of  o;^— 3x^—42.-^—40=0  is  — ^(S-f-y^^M) ; 
find  the  other  roots.     Ans.  — A(3— |/— 31),  4,  and  — 1. 

4.  Two  roots  of  a^— 10a:*H- 29^3— 10a:2—62x-f  60=0 
are  3  and  ^2 ;  find  the  other  roots.     A.  — 1/2,  2,  and  5. 

402.  Proposition  VIII— Descartes'  Rule  of  the 
Signs. — No  equation  can  have  a  greater  number  of  POSITIVE 
roots  than  there  are  VARIATIONS  of  sign;  nor  a  greater  num- 
ber 0/ NEGATIVE  roots  than  there  are  PERMANENCES  of  sign. 

In  the  equation  x — a:^0,  where  the  value  of  x  is  -fa, 
there  is  one  variation,  and  one  positive  root. 

In  the  equation  x-{-a=^^,  where  the  value  of  x  is  — a, 
there  is  one  permanence,  and  one  negative  root. 

In  x^ — (a-\-b~)x-\-ah=.{),  where  the  values  of  x  are  -{-a 
and  -\-b,  there  are  two  variations  and  two  positive  roots. 

In  x'^-]r(a-\-b)x-^ab=z{),  where  the  values  of  x  are  —a, 
and  — 6,  there  are  two  permanences,  and  two  negative  roots. 

In  x^ — X — 12=0,  where  .^=-1-4,  and  — 3,  there  is  one 
variation,  and  one  positive  root,  one  permanence^  and  one 
negative  root. 


360  '     RAY'S  ALGEBRA,  SECOND  BOOK. 

If  we  form  an  equation  of  the  third  degree,  (Art.  397), 
whose  roots  are  -f  2,  -f  3,  -|-4,  we  shall  have  x^—dx^ 
-\-26x — 24=0,  where  there  are  three  variations,  and  three 
positive  roots. 

But  if  we  form  an  equation  whose  roots  are  — 2,  — 3, 
-f-4,  we  shall  have  a?-\-x^ — l^x — 24=0,  where  there  is 
one  variation,  and  one  positive  root,  and  two  permanences, 
and  two  negative  roots. 

To  prove  the  proposition  generally,  let  the  signs  of  the  terms  in 
their  order,  in  any  complete  equation  be 

-j--j-  —  +  —  +  +  -|-5  ^^<^1  Ist  a  new  factor 
X — a=0,  corresponding  to  a  new  positive  root  be  introduced,  the 
signs  in  the  partial  and  final  products  will  be 

+  +-  +  -  +  +  + 

+  - 

+  +  -  +  -  +  +  + 


+  d=-  +  -  +±=i= 


Now,  in  this  product,  it  is  obvious,  that  each  permanence  is  changed 
into  an  ambiguity ;  hence,  the  permanences,  take  the  ambiguous  sign 
as  you  will,  are  not  increased  in  the  final  product;  but  the  number 
of  signs  is  increased  by  one,  and  therefore  the  number  of  variations 
miftt  be  increased  by  one. 

Hence,  the  introduction  of  any  positive  root  introduces  at  least 
one  additional  variation  of  sign. 

Let  us  now  begin  with  the  equation  x — a=^0,  which  contains  one 
positive  root,  and  has  one  variation  of  sign.  Then,  since  every 
additional  positive  root  introduces  at  least  one  additional  variation 
of  sign,  the  number  of  positive  roots  can  never  exceed  the  number  of  vari- 
ations of  sign. 

Again,  if  we  change  the  signs  of  the  alternate  terms,  the  roots 
will  be  changed  from  positive  to  negative,  and,  conversely,  (Art. 
400),  the  permanences  and  variations,  in  the  proposed  equation, 
will  be  interchanged. 

But  since  the  changed  equation  can  not  have  a  greater  number 
of  positive  roots  than  there  are  variations  of  sign,  the  proposed 
equation  can  not  have  a  greater  number  of  negative  roots  than  there  are 
permanences  of  sign. 


GENERAL  THEORY  OF  EQUATIONS.       361 

Corollary  I. — In  an  equation  of  the  m'^*  degree,  since 
the  sum  of*  the  variations  and  permanences  is  equal  to  m, 
the  number  of  real  roots  in  any  equation  can  not  be  greater 
than  its  degree. 

Corollary  II. — If  the  number  of  real  roots  be  less  than 
the  degree  of  the  equation,  the  remaining  roots  are  im- 
aginar?/. 

Take,  for  example,  the  equation 

x'-{-16=0,  or  a)2±0:c+16=i0. 

Taking  the  upper  sign,  there  are  no  variations ;  hence,  there  is 
no  positive  root :  taking  the  lower  sign,  there  are  no  permanences  ; 
hence,  there  is  no  negative  root.  But  the  equation  has  two  roots 
(Art.  396);  they  must,  therefore,  both  be  imaginary. 

Take,  again,  the  cubic  equation 

a:3_|_Bx+C=0,   or  x'dtzOx'-\-Bx-^C=0. 

Reasoning  as  before,  we  find  that  there  can  be  but  one  real  root, 
which  is  negative.  Therefore,  the  other  two  roots  must  be  im- 
aginary. 

403.  Proposition  IX. — If  two  nnmbers.  when  suhsiituted 
for  the  unknown  quantity  in  an  equation^  give  results  affected 
with  different  signs,  one  root,  at  least,  of  this  equation  lies 
between  these  numbers. 

Let  the  equation,  for  example,  be  x^ — x^-{-x — 8=i0. 

If  we  substitute  2  for  x  in  this  equation,  the  result  is  — 2;  and 
if  we  substitute  3  for  x,  the  result  is  -|-13.     It  is  required  to  show 
that  there  must  be  one  real  root,  at  least,  between  2  and  3. 
The  equation  may  evidently  be  written  thus, 

(^x3-\-x)—{x^-]-8)=:0. 
Now,  in  substituting  2  for  ic,  x^-^x=10,  and  x^-\-8=12; 

Therefore,  X^-\-X<:^X^-\-8; 
Also,  in  substituting  3  for  X,  x^-\  x—SO,  and  2:24-8— 17; 
Therefore,  x^-\-xyx^-^8. 
2d  Bk.         31* 


362  RAY'S  ALGEBRA,  SECOND  BOOK. 

Now,  both  members  of  the  inequality  increase  while  x  increases, 
but  the  first  increases  more  rapidly  than  the  second,  since  when 
a:=:2,  it  is  le8&  than  the  second,  but  when  a::=3,  it  is  greater.  Con- 
sequently, for  some  value  of  x  between  2  and  3,  we  must  have 
x'^-\-x^zz.x'^-\-%  and  this  value  of  x  is,  therefore,  a  real  root. 

In  general,  suppose  X=zO  to  be  a  polynomial  equation  involv- 
ing ic,  and  that  'p  and  Q',  when  substituted  for  a:,  give  results  with 
contrary  signs.  Let  P  be  the  sum  of  the  positive,  and  N  the  sum 
of  the  negative  terms.  When  a;=p,  let  P — N  be  negative,  or  P<^N; 
and  when  x=g,  let  P — N  be  positive,  or  P>N. 

Now,  there  must  be  some  value  of  x  between  p  and  g,  which 
renders  P=zN,  or  satisfies  the  equation  X=0.  This  value  of  X  is, 
therefore,  a  real  root  of  the  equation. 

Corollary. — If  the  difference  between  p  and  q  is  equal 
to  iinity^  it  is  evident  that  we  have  found  the  intngral  part 
of  one  of  the  roots. 

1.  Find  the  integral  part  of  one  value  of  x  in  the  equation 

X*— 42^3  4-3a;2-|-x— 5=0. 

If  x=7:3,  the  value  of  the  expression  is  — 2;  but  if  a;=r:4,  the  value 
is  47.     Hence,  8  is  the  first  figure  of  one  root. 

2.  Required  the  first  figure  of  one  of  the  roots  of  the 
equation  x^ — hx^ — x-|-l=0.  Ans.  5. 


TRANSFORMATION    OF    EQUATIONS. 

404.    The   Transformation  of   an   Equation   is  the 

changing  of  it  into  another  of  the  same  degree,  whose  roots 
shall  have  a  specified  relation  to  the  roots  of  the  given 
equation. 

Thus,  in  a«-j-Aa;«-^+Bx"-2.  ^  ^  ^  _^Tx+V=0  ;     (1) 
if  — y  be   substituted  for  x^  the  equation  will   be   trans- 
formed into  another  whose  roots  are  the  same  as  those  in 
(1),  but  with  contrary  signs,  for  ?/= — x. 

If  -  be  substituted  for  X\  then,  y=-,  and  the  roots  of  the  new 
equation  in  y  will  be  the  reciprocals,  of  those  of  equation  (1). 


TRANSFORMATION  OF  EQUATIONS.  363 

4:OS.  Proposition  I. —  To  transform  an  equation  into  one 
whose  roots  are  the  roots  of  the  given  equation  multiplied  or 
divided  hy  any  given  quantity. 

Let  a,  Z>,  c,  etc.,  be  the  roots  of  the  equation 

a;«+Aa:"-i+Bx«-2.   .   .   .   -fTa:+yz=0.      (1). 

Assume  yz=hx^  or  a:=j.     Substituting  this  value  for  a:,  in  (1), 

yn  yu-i  yn-2  ^y 

|+A|^+Bf^ +|+^=«' 

Hence,  2/'*+AA-2/^i+B%"-2.    .     .     .     4-TA-"-i|/-f-/i:"Vz^0. 

Since  y^zlcx^  the  roots  of  this  equation  are  ka^   "kb,  7tC,  etc. 

It  is  evident  that  this  equation  may  be  derived  from  (Ij;  or  that 
the  transfox-mation  of  (1)  is  effected,  by  multiplying  the  successive 
terms  by  1,   A",   A"-,   Z'^,  etc.,  and  changing  X  into  y. 

In  the  case  of  division,  assume  2/— y,  ov  x^^Tcy^  aud  substitute. 

Corollary. — By  this  transformation  an  equation  may  be 
cleared  of  fractions,  or  the  cotifl&cient  of  the  first  term  may 
be  made  unity. 

1.  Let  it  be  required  to  transform  the  equation 

into  one  which  is  clear  of  fractions,  and  which  has  unity 
for  the  coefficient  of  the  term  containing  the  highest 
power  of  X. 

Multiplying  by  6,  Qx^-\-^px'^+2qx-{-Qr=0. 

Putting  y^Qx,  or  x=yy,    6^+3p^V2g^-f  6r=0; 

Multiplying  by  62,   ?/3-f  3j92/2+12g?/+216r=.0. 

2.  Find  the  equation  whose  roots  are  each  3  times  those 
of  the  equation  a;*-]- 7x^—40^+ 3=0. 

Ans.  y+63/— 108y+243==0., 


364 


RAY'S  ALGEBRA,  SECOND  BOOK. 


8.  Find  the  equation  whose  roots  are  each  5  times  those 
of  the  equation  x*-\-2x^ — ^x — 1=0. 

Ans.  y-flO/_8753/— 625r=0. 

4.  What  is  the  equation  whose  roots  are  each  4  of  those 
of  re'— 3x2+4x+10=0  ?        Ans.  4/— 6^^+4^+5=0. 

5.  Transform  eq.  x^ — 2a;^+Ja; — 10=0,  into  one  having 
integral  coefficients.  Ans.  y — %^+3^ — 270=0. 

4:06.  Proposition  II. — To  transform  an  equation  into 
one  whose  roots  are  greater  or  less  hy  any  given  quantity  than, 
the  corresponding  roots  of  the  proposed  equation. 

Let  £c"+Ax"-^+Ba;"-2 _|_Ta;+V=0,  be  an  equa- 
tion whose  roots  are  a,  6,  c,  etc. 

The  relation  between  X  and  y  will  be  expressed  by  the  equation 
yz=xdtir.  As  the  principle  is  the  same  in  both  cases,  let  y^^X — r, 
or  x=:zy-\-r.     Substituting  y-\-r  for  x,  we  have 

(2/+r)»+A(2/+r)«-i+B(2/+r)«-2 _|_T(^+r)+V=.0. 

Developing  the  different  powers  of  y-\^r  by  the  Binomial  Theorem, 
and  arranging  the  terms,  we  have 


y'^j^nr 
+  A 


n-l  I  ^(^-1) 


y.2 

1-2 

+(n— l)Ar 


y. 


+7^ 

+Ar«-i 
+Br'^2 


-fTr 


Now,  since  y=x — r,  the  values  of  y  in  this  equation  are  a — r, 
b — r,  c — r,  etc. 


40T.  Corollary. — By  means  of  the  preceding  transfor- 
mation, we  may  remove  any  intermediate  term  of  an  equar 
tion.  Thus,  to  transform  an  equation  into  one  which  shall 
want  the  second  term,  r  must  be  assumed  so  that  nr+ A=0. 


TRANSFORMATION  OF  EQUATIONS.  365 

To  take  away  the  third  term,  put  ^n^n — l)r^-\-(n — l)Ar-|- 

1.  Transform  the  equation  a:^ — 'Jx-\-^=^0  into  another 
whose  roots  shall  be  less  by  one  than  the  corresponding 
roots  of  this  equation.  Ans.  y-j-3y — 4i/-\-lz=^0. 

2.  Find  the  equation  whose  roots  -are  less  by  3  than 
those  of  the  equation  x*~-dx^— lbx^-\- 4^9 x— 12=^0. 

Ans.  y-f-9/-f  12y— 14y=0. 

3.  Transform  eq.  x^ — 6x'^-\-Sx — 2=0  into  another  whose 
second  term  shall  be  absent.  Ans.  y — 4y — 2=0. 

408.  There  is  an  easier  and  more  elegant  method  of 
transformation,  which  we  will  now  proceed  to  explain. 

Let  the  proposed  equation  be 

Aa;*-fBa;3-fCa;2-[-Da;+E=0,  (1) 

and  let  it  be  required  to  transform  it  into  another,  whose 
roots  shall  be  less  by  r;  then,  i/=x — r  and  x=i/-\-r. 

By  substituting  y-\-r,  instead  of  x,  we  have 

A(2/+^)'+B(2/+r)-^+C(2/-i-r)2+D(2/-f?*)+E-0. 

By  developing  the  powers  of  2/+^?  and  arranging,  as  in  Art.  406, 
the  transformed  equation  will  take  the  form 

A2/<+Bi?/-'5+C,2/2+I>i2/+Ei=0,  (2) 

where  A  is  evidently  the  same  as  in  (1),  while  Bj,  Ci,  Di,  and  Ei, 
are  unknown  quantities  to  be  determined.  For  y,  substitute  its 
value  x — r,  and  equation  (2)  becomes 

A(a:— r)4-[-Bi(a:— r)3-fCi(rc— r)2+Di(a:— r)+Ei=:0.     (3) 

Now,  since  the  values  of  X  are  the  same  in  (1)  and  (3),  these 
equations  are  identical.  Hence,  any  operation  may  be  performed 
on  (1)  or  (3)  with  the  same  result. 

Now,  as  the  object  is  to  obtain  the  values  of  Bj,  Cj,  etc.,  let  (3) 
or  (1)  be  divided  by  X—r,  and  the  quotient  will  be 

A(a;— r)3+Bi  (x—r)^-\-C^  (a:— r)+Di, 
with  the  remainder  E^ ;  hence,  E^  is  determined. 


366 

Divide  this  quotient  by  X — r,  and  the  next  quotient  will  be 
A(:r_r)2+Bi(a:-r)+Ci, 
with  a  remainder  D^;  hence,  Di  is  determined. 

Continuing  the  division  by  X — r,  we  obtain  Cj  and  Bj,  and  thus 
find  all  the  coefficients  of  equation  (2). 

To  illustrate,  let  us  now  solve  Ex.  1,  Art.  407,  by  this  method. 

Transform  the  equation  ic' — 7cc-|-7=0  into  another, 
whose  roots  shall  be  less  by  1  than  the  corresponding 
roots  of  this  equation. 

Here,  2/=^ — 1,  and  we  proceed  to  divide  the  proposed  equation 
and  the  successive  quotients  by  X — 1.  The  successive  remainders 
will  be  the  coefficients  of  2/  in  the  transformed  equation,  except  that 
of  the  highest  power,  which  will  have  the  same  coefficient  us  the 
highest  power  of  x  in  the  proposed  equation. 


x—\  )x^—ix^  7  (a;2+a:— 6 

X^—X^               Ist  quot. 

+a:2— 7a; 

X^—  X 

rc— l)a;2-fa:— 6(a:-f-2 

X^—X           2d  quot. 

-f2a;— 6 
2a:-2 

— 6a:+7- 
— 6a:+6 

iBtrem.  =+1 

2d  rem.    = — 4 

a:— l)a:-f2(l,  ndquot. 
x-\ 

3d  rem.    r-^-j-o 

Since  the  successive  remainders  are  -f-3,  — 4,  and  -fl,  we  have 
A=l,  Bi3=-}-3,  Ci= — 4,  and  D]=-)-l.  Hence,  the  transformed 
equation  is  y'^^Zy^—\y-\-\=:S). 

This  method  of  transforming  an  equation  may  be  greatly 
shortened  by  Horner's  Synthetic  Method  of  Division,  which 
we  shall  now  proceed  to  explain. 

409.  Synthetic  Division.— This  may  be  considered  as 
an  abridgment  of  the  method  of  division  by  Detached 
Coefficients  (Art.  77).  To  explain  the  process,  we  shall 
first  divide  5x* — \2x^-\-Zx^-\-4iX — 5  by  x — 2,  by  detached 
coefficients. 


TRANSFORMATION  OF  EQUATIONS.  367 

By   changing    the    sign    of  Divisor.                                Quotient, 

the  second  term  of  the  divi-  1—2)5—12+3+4—5(5—2—1+2, 

8or,  and  adding  each   partial  5—10           or  bx^—2x'^—X+2. 

product,  except  the  first  term,  —2+3 

which,  being  always  the  same  — 2+4 

as  the  first  term  of  each  divi-  -^  i  ^ 

dend,  may  be  omitted^  the  op-  l_|-2 

eration   may  be    represented  ■        '~t~ 

as  in  the  margin  below:  ^^      . 


Let  it  be  observed  that  the     1+2)5—12+3+4—5(5—2—1+2 
figures  over  the  stars  are  the  •*+10 

coefficients     of    the     several  9  ,  q 

terms  of  the  quotient;    also,  .:;. * 

that  it  is  unnecessary  to  bring 
down  the  several  terms  of  the 
dividend. 

Hence,   the   last    operation  ^^     ^ 

may   be   represented   as    fol-    '  "~^'* 

lows :  —1 


1+4 

*_2 


+2)5—12+3+4—5 
+10-4—2+4 

_  2-1+2—1 

In  this  operation,  5  is  the  first  term  of  the  quotient,  +10  is  the 
product  of  +2,  the  divisor,  by  5;  the  sum  of  +10  and  — 12  gives 
— 2,  the  second  term  of  the  quotient;  +2X — 2= — 4,  and  —4  and 
+3  gives  — 1,  the  third  term  of  the  quotient,  and  so  on.  The  last 
term,  — 1,  is  the  remainder. 

Supplying  the  powers  of  re,  the  quotient  is  bx^ — 2x'^ — ^+2,  with 
a  remainder  — 1. 

A  similar  method  may  be  used  when  the  divisor  contains  three 
terms,  but  the  process  is  more  complicated. 

If  the  coefficient  of  the  first  term  of  the  divisor  is  not  unity,  it 
may  be  made  unity  by  dividing  both  dividend  and  divisor  by  tiie 
coefficient  of  the  first  term  of  the  divisor. 

If  any  term  is  wanting,  its  place  must  be  supplied  with  a  zero. 

410.  In  application  of  these  principles, 


368  RAY'S  ALGEBRA,  SECOND  BOOK. 

1,  Let  it  be  required  to  find  the  equation  whose  roots 
are  less  by  1  than  those  of  the  equation  x^ — ta:-f  7. 

Since  the  second  term  is  wanting,  its  place  must  be  supplied 
with  0.     The  divisor  is  X — 1;  hence,  we  divide  by  -j-l- 

OPERATION    BY    SYNTHETIC    DIVISION. 

-fl)     1     ±0    —7     +7 
+1     +1    -6 

+1     +2 

+2    —4  .-.  —4=  2d  R. 

-f  3  .-.  -f  3=  3^  R. 
Hence,  the  required  coefficients  are  1,  -(-3,  — 4,  and  -\-l. 
.'.  y^-\-^y'^ — 4y-|-l=0  is  the  transformed  equation  required. 

2.  Transform  the  equation  5a-*-j-28a;3-f  51a;2-f-32x— 1^=0, 
into  another  having  its  roots  greater  by  2  than  those  of 
the  given  equation. 

Here,  7/i=a;-f-2;  hence,  we  divide  by  —2,  thus, 

—2)    5     +28    +51     +32    —1 
_10    —36    —30    —4 


+18     +15    +2    — 5  .-.  — 5=  1*' R. 
_10    —16     +  2 


+  8    —  1     +  4  .-.  +4=  2'i  R. 
-10    +  4 


_  2     +  3  .-.  +3=  S'^  R. 
—10 


—12  .-.  -12=  4'A  R. 

Hence,  A=5,    Bi=— 12,    Ci=+3,    D,=+4,  and  E,=— 5  .-.  the 
transformed  equation  is  5?/'' — l'2.y^-{-^y'^-\-Ay—b=:^0. 

3.  Find  the  equation  whose  roots  are  less  by  1.7  than 
those  of  the  equation  u? — 2x'^-\-Zx — 4=0. 


TRANSFORMATION  OF  EQUATIONS.       369 

If  we  transform  this  equation  into  another  whose  roots  are  less 
by  1,  the  resulting  equation  is  2/^+2/^+2^/ — 2=0.  We  may  then 
transform  this  into  another  whose  roots  are  less  by  .7,  or  the  whole 
operation  may  be  performed  at  once,  as  follows : 

+1.7)     1 


—2 

+1.7 

+3 
.51 

-4 

+4.233 

—  .3 

+1.7 

+1.4 
+1.7 

+2.49 

+2.38 

+4.87  .• 

+  .233  .-.  +.233=  1st  R 
'.  +4.87=  2d  R. 

+3.1   .-.  3.1=  Sd  R. 
Hence,  the  equation  is  2/H3.1^2_^4.872/+.233=0. 

4.  Find  the  equation  whose  roots  are  each  less  by  3  than 
the  roots  of  a^—2lx—S6=0.        Ans.  ^3_|_9y_90^0. 

5.  Required  the  equation  whose  roots  are  less  by  5  than 
those  of  the  equation  x*—lSx^—S2x''-\-11x-\-9=0. 

Ans.  y+2/-152/—1158y— 2331=^0. 

6.  Required  the  equation  whose  roots  are  less  by  1.2  than 
those  of  the  equation  x^—6x*-\-lAa^-^1. 92x^—11. SI 2x 
—.79232=0.  Ans.  f—1f-\-27/—S=0. 

Transform  the  following  equations  into  others  wanting 
the  2d  term.     (See  Art.  407.) 

7.  x'—6x'-{-7x—2=0,  Ans.  y— 5y— 4=0. 

8.  x'—6x'-\-12x-{-19=^0.  Ans.  y»+27=0. 

Transform  the  following  equations  into  others  wanting 
the  3d  term  : 

9.  a^—Qx'-j-9x—20=0. 

Ans.  y+3y— 20=0,  or  /— 3/— 16=0. 

10.  x^— 4x2+ 5a:— 2=0. 

Ans.  y^ — ^y^=0,  or  i/^4-y^ — 27=^* 


370 


RAY'S  ALGEBRA,  SECOND  BOOK. 


411.  Proposition  III. —  To  determine  the  law  of  Derived 
Polynomials. 

Let  X  represent  the  general  equation  of  the  n^^  degree ; 
that  is, 

X=:a;«+Aa:«-i+Bx"-2.  .  .  .  -fTx+Vr^O. 

If  we   substitute  x-\-h  for  a:,  and  put  Xi  to  represent  the  new 
value  of  X,  we  have 

Xi=(a:+/t)"+A(a;+/i)''-i+B(a;+7i)«-2-f,  etc., 

and  if  we  expand  the  different  powers  of  X-\-h  by  the  binomial 
theorem,  we  have  Xi= 


+Aa:»-J 
+B:r"-2 
-]-,  etc. 


+  (/i— l)Aa:«-2 
+(7i— 2)Ba;'*-3 
-f-,  etc. 


I     /l2 

w(n— l)a:"-2    ^-7^+,  etc. 


+(w— l)(n— 2)Aa;"-3 
+(n— 2)(n— 3)Ba;'*-4 


etc. 


But  the  first  vertical  column  is  the  same  as  the  original  equation, 
and  if  we  put  X',  X'',  X"\  etc.,  to  represent  the  succeeding  col- 
umns, we  have 

X  =ra:'*+Aa:»-i+Ba:"-2-f-,  etc., 
X'  =na;"-i + {n—1 ) Aa:"-2+ (n— 2)Ba;"-34-,  etc., 
X"=n{n—\  )x"-2-^  (n—1 )  (n— 2)  Ax"-3-|-,  etc.. 
Etc.,  etc. 

By  substituting  these  in  the  development  of  Xj,  we  have 
Xi=X+X'Hj^/i2-f  ^-^713+,  etc. 


The  expressions  X',  X",  X'",  etc.,  are  called  derived 
polynomials  of  X,  or  derived  functions  of  X.  X'  is  called 
the  first  derived  polynomial  of  X,  or  first  derived  function 
of  X  ;  X"  is  called  the  second^  X!"  the  iliird,  and  so  on. 

It  is  easily  seen  that  X'  may  be  derived  from  X,  X" 
from  X',  etc.,  hy  mnltiplylng  each  term  hy  the  exponent  of  x 
ii^that  term^  and  diminishing  the  exponent  hy  unity. 


TRANSFORMATION  OF  EQUATIONS.  371 

412.  Corollary.— If  we  transpose  X,  we  have  Xj— X 

X" 
=X.'h-{-z; — 7^^^+j  6tc.     Now,  it  is  evident  that  h  may  be 

X'' 
taken  so  small  that   the  sign  of  the  sum  X'A-f- — ^^^-f, 

etc.,  will  be  the  same  as  the  sign  of  the  first  term  X'A. 

For,  since  X'h+^X"h^-]-,  etc.,  =A(X'+^X"/i+,  etc.),  if  h  be 
taken  so  small,  that  ^X''h-{- ^X'^h'^-^,  etc.,  becomes  less  than 
X'  (their  magnitudes  alone  being  considered),  the  sign  of  the 
sum  of  these  two  expressions  must  be  the  same  as  the  sign  of  the 
greater  X^ 

413.  By  comparing  tlie  transformed  equation  in  Art. 
406,  with  the  development  of  Xj  in  Art.  411,  it  is  easily 
seen  that  Xi  may  be  considered  the  transformed  equation, 
1/  corresponding  to  x,  and  r  to  h. 

Hence,  the  tranformed  equation  may  be  obtained  by  sub- 
stituting the  values  of  X,  X',  etc.,  in  the  development 
of  Xj.     As  an  example. 

Let  it  be  required  to  find  the  equation  whose  roots  are 
less  by  1  than  those  of  the  equation  x^ — ^x~{-7=^0. 

Here,     ...         X  =a:3.-7a:+7,  X"'=6, 

X'  =3a:2— 7,  Xiv=0. 

X"=^6x, 

Observing  that  h=zl,  and  substituting  these  values  in  the  equa- 

v//  v/// 

tion  Xi=^X-fX^^+:p-^/t2-f  h^^,  etc.,  we  have  X^={x^—7x 

1  n 

-f-7)-K3a;2  — 7)l+(6a;).,— ^-f  =x^j^Zx:^—ix^l,  in  which 

ihe  value  of  a;  is  equal  to  that  of  x  in  the  given  equation  diminished 

^yi. 

By  this  method,  solve  the  examples  in  Art.  410, 


372  RAY'S  ALGEBRA,  SECOND  BOOK. 

EQUAL     ROOTS. 
414.    To  determine  the  equal  roots  of  an  equation. 

We  have  already  seen  (Art.  396,  Rem.)  that  an  equa- 
tion may  have  two  or  more  of  its  roots  equal  to  each  other. 
"We  now  propose  to  determine  when  an  equation  has  equal 
roots,  and  how  to  find  them. 

If  we  take  the  equation  (x — 2)^=0  (1),  its  first  derived 
polynomial  is  Z(x — 2)^=0. 

Hence,  we  see  that  if  any  equation  contains  the  same  factor  taken 
three  times,  its  first  derived  polynomial  will  contain  the  same  factor 
taken  twice;  this  last  factor  is,  therefore,  a  common  divisor  of  the 
given  equation,  and  its  first  derived  polynomial. 

In  general,  if  we  have  an  equation  X=0,  containing  the  factors 
{x — ay^{x — 6)",  its  first  derived  polynomial  will  contain  the  fac- 
tors m[x — a)'^~'^n{X — 6)"-l ;  that  is,  the  greatest  common  divisor 
of  the  given  equation,  and  its  first  derived  polynomial,  will  be 
[x — a)'"~i(a;— 6)"-\  and  the  given  equation  will  have  m  roots,  each 
equal  to  a,  and  n  roots,  each  equal  to  b 

Therefore,  to  determine  whether  an  equation  has  equal  roots, 

Find  the  greatest  commori  divisor  between  the  equation  and 
its  first  derived  'polynomial.  If  there  is  no  common  divisor^ 
the  equation  has  no  equal  roots. 

If  the  G.C.D.  contains  a  factor  of  the  form  x — a,  then  it 
has  two  roots  equal  to  a;  if  it  contains  a  factor  of  the 
form  (x — ay  it  has  three  roots  equal  to  a,  and  so  on. 

If  it  has  a  factor  of  the  form  {x — a){x — 6)  it  has  two 
Voots  equal  to  a,  and  two  equal  to  6,  and  so  on. 

1.  Given  the  equation  x^ — x^ — 8.x-|-12=0,  to  determine 
whether  it  has  equal  roots,  and  if  so,  to  find  them. 

We  have  for  the  first  derived  polynomial  (Art.  411),  Sa:^— 2a:— 8. 

The  G.C.D.  of  this  and  the  given  equation  (Art.  108)  is  x—2. 
Hence,  iC— 2=0,  and  a:=-|-2.  Therefore,  the  equation  has  two  roots 
equal  to  2. 


LIMITS  OF  THE  ROOTS  OF  EQUATIONS.  373 

Now,  since  the  equation  has  two  roots  equal  to  2,  it  must  be  divis- 
ible by  {x—2){x—2),  or  (x-2)2.     (Art.  395).     Whence, 

a;3_x2_8a:-fl2==(a:— 2)2(a:+3)=0,  and  a:+3=.0,  or  x=— 3. 

Hence,  when  an  equation  contains  other  roots  besides  the  equal 
roots,  the  degree  of  the  equation  may  be  depressed  by  division,  and 
the  unequal  roots  found  by  other  methods. 

The  following  equations  have  equal  roots  ;  find  all  the 
roots. 

2.  a:»— 2x2— 15a;4- 36=0.      .     ,     _      Ans.  3,  3,  —4. 
8.  x'—^x'^^x-\-\2=0.  .     .     .  Ans.  2,  2,  —1,  —3. 

4.  a;*— 6x3+12x2— 10x+3=0.  Ans.  1,  1,  1,  3. 

5.  a;*_7x3-f-9x2+27x— 54=0.  Ans.  x=3,  3,  3,  —2. 

6.  x*+2x3— 3x2— 4x+4=0.  Ans. —2,— 2^-1-1, +1. 
Y.  X*— 12x^4-50x2— 84x+49=0.  A.  3±i/2,  3±|/2. 

8.  x^— 2x*+3x3— 7x2+8x— 3=0.  

Ans.  1,  1,  1,  _i±iy^— 11. 

9.  x''-}-3x5— 6x*— 6x3-f  9x2-f  3x— 4=0. 

Ans.  1,  1,  1,  —1,  —1,  —4. 

Suggestion. — In  the  solution  of  equations  of  high  degree, 
the  principles  above  explained  may  be  extended.  Thus,  in  the  last 
example,  the  G.C.D.  is  X^ — X^ — x-f-1.  Proceeding,  we  may,  1st,  find 
the  common  measure  of  this  and  its  first  derived  polynomial,  and 
thus  resolve  into  factors;  or,  2d,  find  the  G.C.D.  of  the  first  and 
second  derived  polynomials.  If  it  is  of  the  form  X — a,  one  of  the 
factors  of  the  original  equation  will  evidently  be  (X — a)^,  etc. 

By  the  1st  method,  we  find  x^— x2— x-fl=(x— l)(x2— l)=(x— 1)2 
(x+1);  by  the  2d,  (x— 1)3  is  a  factor  of  the  original  equation; 
hence,  (x — 1)2  is  a  factor  of  x3_x2— x-(-l. 


LIMITS    OF    THE    ROOTS    OF    EQUATIONS. 

415.  Limits  to  a  Root  of  an  Equation  are  any  two 

numbers  between  which  that  root  lies. 

A  Superior  Limit  to   the   positive   roots   is  a  number 
numerically  greater  than  the  greatest  positive  root. 


374  RAY'S  ALGEBRA,  SECOND  BOOK. 

Its  characteristic  is,  that  when  it,  or  any  number  greater 
than  it,  is  substituted  for  x  in  the  equation,  the  result  is 
positive. 

An  Inferior  Limit  to  the  negative  roots,  is  a  number 
numerically  greater  than  the  greatest  negative  root.  The 
substitution  of  it,  or  any  number  greater  than  it,  for  x, 
produces  a  negative  result. 

The  object  of  ascertaining  the  limits  of  the  roots  is  to 
diminish  the  labor  necessary  in  finding  them. 

416.  Proposition  I. —  The  greatest  negative  coefficient^ 
increased  hy  unity ^  is  greater  than  the  greatest  root  of  the 
equation. 

Take  the  general  equation 

^n_|_Ax«-i-|-Bx»-2 j^Tx-\-Y=0, 

and  suppose  A  to  be  the  greatest  negative  coefiicient. 

The  reasoning  will  not  be  affected  if  we  suppose  all  the 
ooefficients  to  be  negative,  and  each  equal  to  A. 

It  is  required  to  find  what  number  substituted  for  x  will 
make  a;»>A(x"~i-f-a;"-2_|_^»-3^  ^  ^  ^   _|_^_|_1). 

x^ 1 

By  Art.  297,   the  sum   in  parenthesis  is  *- — y ;  hence,  we  must 

X — J. 


havea;»>A(|^),  or  a:">. 


A  a:'* 


1      x—V 


But  if  x''=- — -,  we  find  a:=A4-l;  .-.  A+1  substituted  for  x  will 
X — J. 

„      Aa:"        ,                    ^,       „^  Ax"        A 
render  x'= — ,  and,  consequently,  a;"_> ^ ^. 

By  considering  all  the  coefficients  after  the  first  negative,  we  have 
taken  the  most  unfavorable  case ;  if  any  of  them,  as  B,  were  posi- 
tive, the  quantity  in  parenthesis  would  be  less. 

41T.  Proposition  II. — If  we  take  the  greatest  negatice 
coefficient^  extract  a  root  of  it  whose  index  is  equal  to  the 
number  of  terms  preceding  the  frst  negative  term,  and  in- 
crease it  hy  unify,  the  result  will  he  greater  than  the  greatest 
positive  root  of  the  equation. 


LIMITS  OF  THE  ROOTS  OF  EQUATIONS.  375 

Let  Cx^~^  be  the  first  negative  term,  C  being  the  great- 
est negative  coefficient ;  then,  any  value  of  x  which  makes 

cc">C(x»-'--f-a;»-'-i +^^+1)     (1) 

will  render  the  first  of  the  proposed  equation  >>0,  or 
positive ;  because  this  supposes  all  the  coefficients  after 
C  negative,  and  each  equal  to  the  greatest,  which  is  evi- 
dently the  most  unfavorable  case. 

By  Art.  297,  the  series  in  parenthesis  = ^ — .     Hence, 

^    \      x—1      /'           ^    x—1        x—1           ' 
this    inequality   will   be    true   if  a:'*^ r — ,  or  > ^r—; 

or,  by  multiplying  both  members  by  iC— 1,  and  dividing  by  0:"-''+', 
when  (a;-l)a:'--i=C,  or  >C  (2). 

But  x—1  is  <iC,  and  .-.  {x—ly-^<^x''~^  •-.  (2)  will  be  true  if  we 
have  {x—l){x—lY-\ 

Or{x-lY^C,  or  >C; 

OriC — 1={/C,  or  >v  C; 

Orar^l  +  v'C;  or  >l  +  v/C: 

Find  superior  limits  of  the  roots  of  the  following  equa- 
tions : 

1.  x'—bx'-\-^1x'—Sx+S9=0. 

Here,  C=5,  and  r=l  .-.  1+^C=1+5tz=6,  Ans. 

2.  x'-^1x'—12a^—49x'Jro2x—lS=0. 
Here,  l+;:/C^l+ 1^49^1 +  7=8,  Ans. 

3.  o^-'+llx^— 25x— 67=0. 

By  supposing  the  second  term  -fOa;',  we  have  r=3 ; 
hence,  the  limit  is  l-[-^67,  or  6. 

4.  Sa^—2x'—Ux-^4=^0. 
Dividing  by  3,  a:^— |ic2_y^_|_4^0. 
Here,  the  limit  is  l  +  V'  ^^  ^-v 


370  RAY'S  ALGEBRA,  SECO^^D  EOOK. 

418.  To  determine  the  inferior  limit  to  the  negative 
TOots,  change  the  signs  of  the  alternate  terms;  this  will 
change  the  signs  of  the  roots  (Art.  400);  then, 

The  superior  limit  of  the  roots  of  this  equation,  by 
changing  its  sign,  will  be  the  inferior  limit  of  the  roots  of 
the  proposed  equation. 

419.  Proposition  III. — If  the  real  roots  of  an  equation^ 
taken  in  the  order  of  their  magnitudes^  he  a,  b,  c,  d,  etc.^  a 
heing  greater  than  b,  b  greater  than  c,  and  so  on;  then,  if 
a  series  of  numbers,  a',  b',  c',  d',  etc.,  in  which  a'  is  greater 
than  a,  b'  a  number  between  a  and  b,  c'  a  number  between 
b  and  c,  and  so  on,  be  substituted  for  x  in  the  proposed  equa- 
tion, the  results  will  be  alternately  positive  and  negative. 

The  first  member  of  the  proposed  equation  is  equivalent 
to  (x — a)(x — b)(x — c')(x — d) =0. 

Substituting  for  x  the  proposed  series  of  numbers  a^,  b\  &,  etc., 
we  obtain  the  following  results: 

{cf —a){a^ —b){a^ — c){a^ — d),  etc =-f  product,  since  all 

the  factors  are  -|-. 

(J)^ — a){b^ — b){b^—c){b^—d),  etc = —  product,  since  only- 
one  factor  is  — . 

{& — a){& — b){G^ — c){&—d),  etc =+  product,  since  two 

factors  are  — ,  and  the  rest  -\-. 

{d^—a){d^~b){d^ — c){d^—d),  etc =— product,  since  an 

odd  number  of  factors  is  — ,  and  so  on. 

Corollary  1. — If  two  numbers  be  successively  substituted 
for  X,  in  any  equation,  and  give  results  with  contrary  signs, 
there  must  be  one,  three,  five,  or  some  odd  number  of  roots 
between  these  numbers. 

Corollary  2. — If  two  numbers,  substituted  for  x,  give 
results  with  the  same  sign,  then  between  these  numbers 
there  must  be  two,  four,  or  some  even  number  of  real  roots, 
or  no  roots  at  all. 


THEOREM  OF  STURM.  377 

Corollary  3. — If  a  quantity  2',  and  every  quantity  greater 
than  g',  render  the  results  continually  positive,  q  is  greater 
than  the  greatest  root  of  the  equation. 

Corollary  4. — Hence,  if  the  signs  of  the  alternate  terms 
be  changed,  and  if  p,  and  every  quantity  greater  than  p^ 
renders  the  result  positive,  then  — p  is  less  than  the  least 
root  of  the  equation. 

Illustration. — If  we  form  the  equation  whose  roots  are  5,  2,  and 
— 3,  the  result  is  x^ — 4iX^ — lla;+30=0.  Now,  if  we  substitute  any 
number  whatever  for  x^  greater  than  5,  the  result  is  positive.  If  we 
put  a;=:5,  the  result  is  zero,  as  it  should  be. 

If  we  substitute  for  rc,  any  number  less  than  5,  and  greater  than 
2,  the  result  is  negative.     Putting  a:— 2,  the  result  is  zero. 

Substituting  for  rc,  any  number  less  than  2,  and  greater  than  — 3, 
the  result  is  positive.     Substituting  — 3,  it  is  zero. 

Substituting  a  number  less  than  — 3,  the  result  is  negative. 

From  Cors.  3  and  4,  it  is  easy  to  find  when  we  have  passed  all  the 
real  roots,  either  in  the  ascending  or  descending  scale. 


STURM'S     THEOREM. 

4L20»  To  find  the  number  of  real  and  imaginary  roots 
of  an  equation. 

In  1584,  M.  Sturm  gained  the  mathematical  prize  of 
the  French  Academy  of  Sciences,  by  the  discovery  of  a 
beautiful  theorem,  by  means  of  which  the  number  and  sit- 
iiaiion  of  all  the  real  roots  of  an  equation  can,  with  cer- 
tainty, be  determined.  This  theorem  we  shall  now  proceed 
to  explain. 

Let  X=rc«-|-Aa;«-i-j-Ba:'»-2 _|_Tx+V=rO,  be 

any  equation  of  the  ?i'^  degree,  containing  no  equal  roots  ; 
for  if  the  given  equation  contains  equal  roots,  these  may 
be  found  (Art.  414),  and  its  degree  diminished  by  di- 
vision. 

2d  Bk.  32 


378  RAY'S  ALGEBRA,  SECOND  BOOK. 

Let  the  first  derived  function  of  X  (Art.  411)  be  denoted  by  Xj. 

Divide  X  by  Xj  until  the  remainder  Xi)X     (Qi 

is  of  a  lower  degree  with  respect  to   x  XjQj 

than  the  divisor,  and  call  this  remain-  Z^     ^  r\           -^^ 
der  — X^;   that  is,  let  the  remainder, 

with  its  sign  changed,  be  denoted  by  Xo.  ^2)^1       (Q2 

Divide  Xj  by  X2  in  the  same  man-  X0Q2 

ner,  and  so  on,  as  in  the  margin,  de-  x, X  Qo=— X 

noting  the  successive  remainders,  with 

their  signs   changed^    by    X3,    X4,    etc.,  ^Z)^2     (vs 

until  we  arrive  at  a  remainder  which  ^sQ.i 


does  not  contain  rr,  which  must  always  X9 XoQozzr X4 

happen,  since  the  equation  having  no 

equal  roots,  there  can  be  no  factor  containing  X.  common  to  the 
equation  and  its  first  derived  function.  Let  this  i-emainder,  having 
its  sign  changed,  be  called  X^^j. 

In  these  divisions,  we  may,  to  avoid  fractions,  either  multiply  or 
divide  the  dividends  and  divisors  by  tiny  positive  number,  as  this  will 
not  affect  the  signs  of  the  functions  X,  Xj,  Xo,  etc. 

By  this  operation,  we  obtain  the  series  of  quantities 

X,  Xi,  X2,  Xo.     .    .    .    Xh-1  (1). 

Each  member  of  "this  series  is  of  a  lower  degree  with  respect  to 
X  than  the  preceding,  and  the  last  does  not  contain  X.  Call  X  the 
primitive  function,  and  Xi,  X2,  etc.,  auxiliary  functions. 

431.  Lemma  I. —  Two  consecutive  functions^  Xj,  X2, /or 
example^  can  not  both  vanish  for  the  same  value  of  x. 

From  the  process  by  which  Xi,  X2,  etc.,  are  obtained,  we  have  the 
following  equations : 

X    =X,Qi-X2 (1) 

Xi    =X2Q2-X3 (2) 

X2  =X3Q3-X, (3) 

X,_i=X,Q.-X,+i yr). 

If  possible,  let  Xi— 0,  and  X2=0;  then,  by  eq.  (2)  we  have 
Xo=0;  hence,  by  eq.  (3)  we  have  X4=0;  and  proceeding  in  the 
same  way,  we  shall  find  Xj^O,  Xg^O,  and  finally  X^i=0.  But 
this  is  impossible,  since  X^-f  1  does  not  contain  iC,  and  therefore  can 
not  vanish  for  any  value  of  x. 


THEOREM  OF  STURM.  379 

4t22*  Lemma  II. — If  one  of  the  auxiliary  functions  van- 
ishes for  any  particular  value  of  x,  the  tico  adjacent  functions 
must  have  contrary  signs  for  the  same  value  of  x. 

Let  us  suppose  that  X3=0,  when  a:=a;  then,  because  X9^X3Q3 
— X4,  and  X3=i0;  therefore,  X2^=— X4;  that  is,  X2  and  X4  have 
contrary  signs. 

4S3.  Lemma  III. — If  any  of  the  auxiliary  functions 
vanishes  when  x=a,  and  h  be  taken  so  small  that  no  root 
of  any  of  the  other  functions  in  series  (1)  lies  between  a — h 
and  a-j-h,  then  will  the  number  of  variations  and  perma- 
nences^ when  a — h  and  a-j-h  are  substituted  for  x  in  this 
series^  be  precisely  the  same. 

Suppose,  for  example,  the  substitution  of  a  for  x  causes  the 
function  X3  to  vanish  ;  then,  by  Art.  421,  neither  of  the  functions 
X2  or  X4  can  vanish  for  the  same  value  of  X]  and  since  when 
X3  vanishes,  Xg  and  X4  have  contrary  signs,  (Art.  422);  therefore, 
the  substitution  of  a  for  x  in  X2,  X3,  X4,  must  give 

^2  >  ^3  ,   X4  ,  or  Xg,  X3,  X4. 

+       0        -  ,         -    0      + 

And  since  h  is  taken  so  small  that  no  root  either  of  X2=0,  or 
X^=iO,  lies  between  a — h  and  CL-\-h^  the  signs  of  these  functions 
will  continue  the  same  whether  we  substitute  a — h  or  a-\-h  for  x 
(Art.  419).  Hence,  whether  we  suppose  X3  to  be  -j-  or  —  by  the 
substitution  of  a — h  and  a-\-h  for  x^  there  will  be  one  variation  and 
one  permanence.     Thus,  we  shall  have  either 

X2  ,  X3  ,  X4  ,  or  Xj  ,  X3  ,  X4. 

+       ±      -  -       ±      + 

So  that  no  alteration  in  the  number  of  variations  and  perma- 
nences can  be  made  in  passing  from  a — h  to  a-\-h. 

4!S4.  Lemma  IV. — If  a  is  a  root  of  the  equation  X=^0, 
then  the  series  of  functions  X,  X^,  Xo,  etc.^  will  lose  one 
variation  of  signs  in  passing  from  a — h  to  a-f-h  ;  h  being 
taken  so  small  that  no  root  of  the  function  Xj=^0  lies  between 
a — h  and  a-i-h. 


380  RAY'S  ALGEBRA,  SECOND  BOOK. 

For  X  substitute  a^li  in  the  equation  X=0,  and  denote  the  result 
by  H.  Also,  put  A,  A^,  A^^  for  the  values  of  X  and  its  derived  func- 
tions when  a-\-}i  is  substituted  for  x\  then  (Art.  411), 

H=A-f  A^/i+JA^^/i2-|-,  etc. 

But,  since  a  is  a  root  of  the  eq.  X=0,  we  shall  have  A=:0,  while 
A^  can  not  be  0,  since  the  eq.  X=iO  has  no  equal  roots.     Hence, 

YL=h'}i-\-\k.''}i^^,  etc.,  =7^(A^+^A^^/i+,  etc)- 

Now,  li  may  be  taken  so  small  that  the  quantity  within  the  paren- 
thesis shall  have  the  same  sign  as  its  first  term  A'',  (since  A^  ex- 
presses the  first  derived  function  of  X,  corresponding  to  X^,  in 
Art.  412);  therefore,  the  sign  of  X,  when  a:=a4-/»,  will  be  the  mme 
as  the  sign  of  X^. 

If  we  substitute  a — }i  for  X  in  the  equation  X:=0,  and  denote  the 
result  by  IF,  we  then  have,  by  changing  li  into  — \  in  the  expres- 
sion for  H, 

\V=~h{iV—\K.''li-^,  etc). 

Now,  it  is  evident  that  for  very  small  values  of  ^■,  the  sign  of  IF 
will  depend  upon  the  first  term  — A^/t,  and,  consequently,  will  be 
contrary  to  that  of  A''.  Hence,  when  X^=a — ^,  there  is  a  variation 
of  signs  in  the  first  two  terms  of  the  series  X,  Xj ;  and  when 
x=^a-\-h^  there  is  a  continuation  of  the  same  sign.  Therefore,  one 
variation  is  lost  in  passing  from  X=z^a—1l  to  a-\-h. 

If  any  of  the  auxiliary  functions  should  vanish  at  the  same  time 
by  making  x=:a^  the  number  of  variations  will  not  be  affected  on 
this  account  (Art.  423),  and  therefore,  one  variation  of  signs  will 
still  be  lost  in  passing  from  a — h  to  a-{-h. 

425.  Sturm's  Theorem. — If  any  hco  numbers^  p  and  q, 
(p  being  less  than  q)  he  substituted  for  x  in  the  series  of 
functions  X,  Xj,  X2,  etc.,  the  substitution  of  p  for  x  giving 
k  variations,  and  that  of  q  for  x,  giving  k'  variations;  then, 
k — k'  will  be  the  exact  number  of  real  roots  of  the  equation 
X=0,  which  lie  between  p  and  q. 

Let  us  suppose  that  —  00  is  substituted  for  x,  and  sup- 
pose that  X  continually  increases  and  passes  through  all 
degrees  of  magnitude  till  it  becomes  0,  and  finally 
reaches  -|-  oc. 


THEOREM  OF  STURM.  381 

Now,  it  is  evident,  that  so  long  as  iC,  -with  its  minus  sign,  is  less 
than  any  of  the  roots  of  Xr^O,  Xi=0,  etc.,  no  alteration  will  take 
place  in  the  signs  of  any  of  these  functions  (Art.  419);  but  when 
X  becomes  equal  to  the  least  root  (with  its  sign)  of  any  of  the 
auxiliary  functions,  although  a  change  may  occur  in  the  sign  of 
this  function,  yet  we  have  seen  (Art.  423)  that  it  is  the  order  only, 
and  not  the  number  of  variations  which  is  affected.  But  when  X  be- 
comes equal  to  any  of  the  roots  of  the  primitive  function,  then  ono 
variation  of  signs  is  always  lost. 

Since,  then,  a  variation  is  always  lost  whenever  the  value  of  x 
passes  through  a  root  of  the  primitive  function  X=0,  and  since  a 
variation  can  not  be  lost  in  any  other  way,  nor  can  one  be  ever  in- 
troduced, it  follows  that  the  excess  of  the  number  of  variations  given 
by  x=p,  above  that  given  by  Xz=q  {p<^q),  is  exactly  equal  to  the 
number  of  real  roots  of  X=0,  which  lie  between  p  and  q. 

Corollary. — If  the  equation  is  of  the  n'^  degree,  and  m 
represents  the  number  of  real  roots,  then  (Art.  396),  the 
number  of  imaginary  roots  will  be  n — m. 

4:!S6«    To  determine  the  number  of  real  roots. 

Substitute  —  oo  and  -j-  go  for  a;  in  the  several  functions, 
since  the  roots  must  all  be  comprised  between  these  limits. 
In  this  case,  the  sign  of  each  function  will  be  that  of  its 
first  term. 

If  we  substitute  0  for  cc,  the  number  of  variations  lost 
from  —  oo  to  0,  will  be  the  number  of  negative  roots ;  and 
from  0  to  -|-  00,  the  number  of  positive  roots. 

4:2T«  To  determine  the  situation  of  each  real  root;  that 
is,  the  figures  between  which  it  lies. 

Substitute  0,  — 1,  — 2,  — 3,  etc.,  for  x,  in  series  (1), 
till  we  find  a  number  which  produces  as  many  variations 
as  ic= —  00  produced.  This  will  be  the  limit  of  the  nega- 
tive roots. 

Substitute  1,  2,  3,  etc.,  till  we  find  a  positive  number  which  gives 
the  same  number  of  variations  that  x=-\-  cx>  does.  This  Avill  be  the 
superior  limit  of  the  positive  roots. 

381 


382  RAY'S  ALGEBRA,  SECOND  BOOK. 

By  observing  where  variations  are  lost,  we  find  the  situation  of 
the  roots.  If  two  or  more  variations  are  lost  between  two  of  the 
substitutions,  take  smaller  numbers,  until  only  one  is  lost.  This  is 
termed  the  separation  of  the  roots. 

1.  Find  the  number  and  situation  of  the  real  roots  of 
the  equation  4x^— 12^2 -fllx— 3=0. 

Here,  we  have  X  =  4a;3— 12a:2-f  Ha:— 3, 

and  (Art.  411)  Xi=12a:^—24x  -fll. 

Multiplying  X  by  3,  to  render  the  first  term  divisible  by  the  first 
term  of  Xj,  and  proceeding  as  in  the  method  of  finding  the  G.C.D., 
(Art.  108),  we  have  for  a  remainder  — 2x-\-2.  Canceling  the  factor 
-|-2,  and  changing  the  signs  (Art.  420),  we  have  X2=x — 1.  Divid- 
ing Xj  by  X2,  we  have  for  a  remainder  — 1 ;  therefore,  X3=-}-l. 
Hence, 

X  =  4a:3— 12a:2+lla;— 3. 

Xi=12a:2— 24a;+ll. 

X2-=    x—1. 

Put  —  Qo  and  -f  oo  for  X,  and  we  have  (Art.  426),  for 
a:=: —  QO,  —  -|-  —  -|-  three  variations,  .-.  k  =3. 
X=-{-  cc,  -\-  -\-  -{-  -\-  no  variation,  .-.  k'=iO. 

.♦.  k — k'=3 — 0=3,  the  number  of  real  roots. 
For  x=Oj  —  -f-  —  -|-)  three  variations  .♦.  k  =3. 

Hence,  there  is  (Art.  426)  no  real  root  between  0  and  —  oo.  This 
we  might  also  have  learned  from  Art.  402,  since  there  is  no  perma- 
nence in  the  proposed  equation. 

It  is  best  to  substitute  integral  numbers  first,  and  afterward  frac- 
tional, if  two  or  more  roots  are  found  to  lie  between  two  whole 
numbers.     Or,  substitute  fractions  at  once,  thus : 

X 

For  x= —  GO  the  signs  are  — 

x=    0 — 

^=+1 - 

^=4-^ 0 

^=+1 + 

ic^-j-l 0       —       0 


X, 

X2 

X3 

+ 

— 

+ 

giving  3  var. 

+ 

— 

+ 

({ 

3     " 

+ 

— 

+ 

(( 

3     " 

+ 

— 

+ 

— 

— 

+ 

« 

2    " 

THEOREM  OF  STURM.  383 

Fora:=-fli^     .....     —      —       +       +     giving  1  var. 

x^+n 0    +    +    + 

x=-^H +    +    +    +     "    0  '« 

a:=+oo +      +      +      +        "       0    " 

The  roots  are  ^,  1,  and  1  j.  If  these  numbers  had  not  been  sub- 
stituted, the  loss  of  one  yariation  in  passing  from  |  to  f ;  one  in 
passing  from  |  to  1|^;  and  one  in  passing  from  1^  to  If,  would  have 
given  the  situation  of  the  roots. 

A  careful  study  of  this  example  will  serve  to  illustrate  the  the- 
orem. Thus,  we  see  that  there  are  three  changes  of  sign  of  the 
primitive  functioUj  two  of  the  first  auxiliary  function,  and  one  of  the 
second. 

Again,  while  no  variation  is  lost  by  the  change  of  sign  of  either 
of  the  auxiliary  functions,  each  change  of  sign  of  the  primitive 
function  occasions  a  loss  of  one  variation. 

2.  How  many  real  roots  has  the  eq.  x' — Sx'^-]-x — 3=0  ? 

Here, X  =  x^—3x^-^x-3 

Xi=3a;2— 6a;H-l 

X2=  x^2 
X3=— 25. 

For  x= —  00  the  signs  are  —  + ,2  variations,  .-.  k=2. 

a;=-|-  00  the  signs  are  +   +   H »  1  variation,     .-.  k'=l. 

.-.  k — k'^2 — 1=^1,  the  number  of  real  roots. 

One  variation  is  lost  in  passing  from  2  to  4  and  X— 0  when 
a:=^3  i  therefore,  the  root  is  -{-3. 

Find  the  number  and  situation  of  the  real  roots  in  each 
of  the  following  equations  : 

3.  x'—2x''—x-{-2=^0.      Ans.  Three.    —1,  +1,  +2. 

4.  Sx'  —  S6x'^46x  —  lb=^0.  Ans.  Three.  One  be- 
tween 0  and  1,  one  between  1  and  2,  one  between  2  and  3. 

5.  rc^— 3a;2— 4x+ll=0.  Ans.  Three.  One  between  —2 
and  — 1,  one  between  1  and  2,  one  between  3  and  4. 

6.  z^^-2x — 5=0.  Ans.  One  between  2  and  3. 


384  RAY'S  ALGEBRA,  SECOND  BOOK. 

Y.  o.-^— 15x— 22rr=0.      Ans.   Three.      One   root  is  —2, 
one  between  — 2|  and  — 2^,  one  between  4  and  5. 

8.  x'—4:x^—8x-i-2S=0.     Ans.  Two.     One  between  2 
and  3,  and  one  between  3  and  4. 

9.  a:*— 2a;3— 7a)2-f-10x+10=0.     Ans.  Four.     The  limits 
are  (-3,  -2);  (0,  -1);  (2,  3);  (2,3). 

10.  a^—10x^-^6x-\-l=0.     Ans.  Five.     The  limits  are 
(-4,  -3);  (-1,  0);  (-1,  0);.(0,  1);  (3,  4). 


XIII.    RESOLUTION    OF    JS^UMERICAL 
EQUATIONS. 

4;28«  In  the  preceding  articles  we  have  demonstrated 
the  most  important  propositions  in  the  theory  of  equations, 
and  in  some  cases  have  shown  how  to  find  their  roots. 

The  general  solution  of  an  equation  higher  than  the 
fourth  degree,  has  never  yet  been  effected.  In  the  prac- 
tical application  of  Algebra,  however,  numerical  equations 
most  frequently  occur ;  and  when  the  roots  of  these  are 
real,  they  can  always  be  found,  either  exactly  or  approxi- 
mately. The  way  for  doing  this  has  been  prepared  in  the 
preceding  articles,  by  finding  the  limits  of  the  roots,  and 
separating  them  from  each  other. 

RATIONAL    ROOTS. 

4SO.  Proposition  I. —  To  determine  the  integral  roots  of 
an  equation. 

If  a  be  an  integral  root  of  the  equation  Aa-*-|  B^'-h^*^' 
-fDa;-fE=0,  we  shall  have  Aa*+Ba'4-Ca'^-f-Da--|-E=0i 

E 

therefore,   — = — Aa' — Ba'* — Qa — D. 


RESOLUTION  OF  NUMERICAL  EQUATIONS  385 

Now,  since  the  second  member  of  the  last  equation  is  evidently  a 

E 
whole  number,  E  is  divisible  by  a.     Put  — =E^;   transpose  D  to  the 

CL 

first  member,  and  divide  by  a\  this  gives 

E'+D 

• = — Aa"— Ba— 0;  .-.  a  is  also  a  divisor  of  E'-f  D. 

E'+D 
Put  — L_z=D,  transpose  C,  and  divide  by  a;   this  gives 

D'-l-C 

'     — — Aa— B:  .-.  a  is  a  divisor  of  D'-fC. 
a 

D'-4-C  C'-l-B 

Again,  put  =C',  transpose  B,  divide  by  a,  and  — ~—= — A. 

C'4-B 
Lastly,  making  =^'i  and  transposing  A,  we  have  B'-f-A=0. 

If,  then,  all  these  conditions  are  satisfied,  a  is  a  root  of  the  pro- 
posed equation;  but  if  any  one  of  them  fails,  a  is  not  a  root.  Hence, 
we  have  the  following 

Rule  for  finding  the  Integral  Roots  of  an  Equation. — 

Divide  the  last  term  of  the  equation  hy  any  of  its  divisors  a, 
and  add  to  the  quotient  the  coefficient  of  the  term  containing  x. 
Divide  this  sum  hy  a,  and  add  to  the  quotient  the  coefficient 
ofx\ 

Proceed  in  this  manner  unto  the  first  term,  and  if  a  be  a 
root,  all  these  quotients  will  be  whole  numbers,  and  the  result 
will  be  0. 

Corollary  1. — It  will  be  easier  to  ascertain  whether 
-\-l  and  — 1  are  roots,  by  trial.  Also,  by  ascertaining  the* 
limits  to  the  positive  and  negative  roots  (Art.  417),  we 
may  reduce  the  number  of  divisors. 

Corollary  2. — If  the  first  coefficient  be  not  unity,  the 
equation  may  have  a  fractional  root.  To  determine  if  this 
be  the  case,  transform  the  equation  into  one  having  its  first 
coefficient  unity  (Art.  405,  Cor.  1),  and  its  roots  integers 
(Art.  399). 

Corollary  3. — "When  all  the  roots  except  two  are  in- 
tegral, divide  the  equation  and  find  the  others  (Art.  396, 
Cor.  1). 

2(1  Bk.         33* 


386  RAY  S  ALGEBRA,  SECOND  BOOK. 

1.  Find  the  rational  roots  of  the  equation 

x'-\-Sx'—4x—12=0. 

Here,  by  Art.  417,  no  positive  root  can  exceed  1-|-^12,  or  4,  and 
the  limit  of  the  negative  roots  is  l-[-3=4. 

It  is  also  found,  by  trial,  that  -fl  and  — 1  are  not  roots. 

We  then  proceed  to  arrange  the  divisors  of  — 12,  among  which  it 
is  possible  to  find  the  roots,  and  proceed  as  follows: 

Last  term  — 12 

Divisors -f  2  ,  -f  3  ,  -f  4  ,  — 2  ,  — 3  ,  -  4 

Quotients _  6  ,  — 4  ,  — 3  ,  -f 6  ,  -f4  ,  -f3 

Add  —4 —10  ,  —8  ,  — 7  ,  +2  ,  — 0  ,  -1 

Quotients —  5  ,       *  ,      *  ,  — 1  ,      0         * 

Add  4-3 —  2  ,  -f  2  ,  +3  , 

Quotients —  1  — 1  ,  — 1  , 

Add  -f  1 0  ,  0,0, 

Since  —8,  — 7,  and  — 1,  are  not  divisible  by  -|-3,  4-4,  and  — 4, 
we  proceed  no  further  with  these  divisors,  as  it  is  evident  that  they 
are  not  roots  of  the  equation.     The  loots  are  -\-2,  — 2,  and  — 3. 

Find  the  roots  of  the  following  equations  : 

2.  a;'— Y-c^-f  36=.0 Ans.  3,  6,  and  —2. 

When  any  term  is  wanting,  as  the  3d  term  in  this  example,  its 
place  must  be  supplied  with  0.  When  there  are  equal  roots,  they 
may  be  found  (Art.  414),  or  having  found  one,  reduce  the  degree  c,f 
the  equation  by  division,  and  proceed  as  before. 


3.  a:'— 6.r;-f  llo;— 6r=0.  .     .     . 

4.  ar'+x2— 4a;— 4=0.       ,     ,     , 

5.  a^—Sx'—46x—72=0.     .     . 

6.  a^s—Sx'— 18.t4-Y2=.0.    .    . 
V.  a;*— 10x»+35x2— 50a;-j-24:=.0 


.      Ans.  1,  2,  8. 

Ans.  2,  -1,  —2. 

Ans.    9,  —2,  —4. 

.  Ans.  3,  6,  — 4. 

Ans.  1,  2,  3,  4. 


HORNER'S  METHOD  OF  APPROXIMATION.  387 

8.  x'-^4x'—x'—lQx—12=0.     Ans.  2,  —1,  —2,  —3 

9.  cc*— 4x3— 19x^+46x-hl20=0.  Ads.  4,  5,  —2,  —3 

10.  x'—2lx'-^Ux-i-120=Q.  Ans.  3,  4,  —2,  —5 

11.  x*+a;3— 29a:'^— 9x+180=:0.    -  Ans.  3,  4,  -3,  -5 

12.  a^—2x'—4x-\-S=.0.  Ans.  2,  2,  —2 

13.  x'Jr^x'—Sx-^lO^r^O.  Ans.  —5,  l^]/"^ 

14.  x*-9x'Jr'^7x'-\-21x—60^0.  Ams.  4,  5,  =±=v  3 

15.  2a:'— 3x2-f-2a;— 3:^.0.  Ans.  |,  ±]/=I 

16.  3a:'— 2:i;2— 6^+4=0.  Ans.  f,  ±|/2 

17.  8x3— 26x^+1  la:-f-10=--0.  Ans.  |,  |(3±|/4l) 

18.  6x*— 25.x'+26a;2+4x— 8=0.  Ans.  2,  2,  f,  —J 

19.  X*— 9.x'4-yx^+ya:— V=0.  Ans.  |,  |,  3±3|/2 

IRRATIONAL   ROOTS— METHODS  OF    APPROXIMATION. 

Having  found  all  the  integral  roots,  we  must  have  re- 
course to  methods  of  approximation,  the  best  of  which  is 
Horner's. 

4SO.  Horner's  Method  of  Approximation. — The  prin- 
ciple of  this  method  de-pends  on  the  successive  transforma- 
tions of  the  given  equation,  by  Synthetic  Division  (Art. 
410),  so  as  to  diminish  its  roots  at  each  step  of  the 
operation. 

Let  the  equation,  one  of  whose  roots  is  to  be  found,  be 
Px«-f  Qx»-i ^T.x+V=0. 

Suppose  a  to  be  the  integral  part  of  the  root  required,  and  r,  S, 
^  ...  the  decimal  digits  taken  in  order,  so  that  x=za-^r-{-8-\-t.  .  . 
Find  a  by  trial,  or  by  Sturm's  theorem,  and  transform  the  equation 
into  one  whose  roots  shall  be  diminished  by  a  (Art.  410). 

Let  P^'^^Q'^"-! ^Ty-\^Y'=^0  he  the  transformed  equa- 
tion; then,  the  value  of  ^  is  the  decimal  r-\-S'\-t ;  and  since 


388  RAY  S  ALGEBRA,  SECOND  BOOK. 

this  root  is  contained  between  0  and  1,  we  may  easily  find  its  first 
digit  r.  Again,  let  the  loots  of  this  equation  be  diminished  by  r, 
and  let  the  transformed  equation  be 

P2:"+Q'^2;"-i _|_T"2;4-V'^=0. 

Now,  the  value  of  z  in  this  equation  is  8-\-t.  .  .  .  ,  and  the  value 
of  s  lies  between  .00  and  .1 ;  that  is,  it  is  either  .00,  .01,  .02,  .  .  .  „ 
or  .09.  But  since  the  figure  8  is  in  the  second  place  of  decimals, 
z^^  z^  .  .  .  .  will  be  small,  and  we  may  generally  find  8  from  the 
equation  T^^2;+V^^=0;  or,  Sr=— Vh-T,  nearly. 

Having  found  S,  diminish  the  roots  of  the  last  equation  by  S,  and 
then  from  the  last  two  terms,  T^^^2;^-(-V^^^,  of  the  resulting  equation, 
•find  t  the  next  decimal  figure,  and  so  on. 

431.  The  absolute  number,  or  last  term,  is  sometimes 
called  the  dividend,  and  the  coefficient  of  the  first  power 
of  the  unknown  quantity,  (as,  T"  or  T"',)  the  incomplete 
or  trial  divisor. 

The  correctness  of  the  values  of  s,  t,  etc.,  obtained  by  means  of 
the  trial  divisor,  will  always  be  verified  in  the  next  operation.  If 
too  great  or  too  small,  the  quotient  figure  must  be  increased  or 
diminished. 

The  accuracy  with  which  each  succeeding  decimal  figure  may  be 
found,  increases  as  the  value  of  the  figure  decreases.  In  general, 
after  finding  three  or  four  decimal  figures,  the  rest  may  be  obtained 
with  suflQcient  accuracy  by  dividing  \^^  by  T^^. 

43!S.  By  changing  the  signs  of  the  alternate  terms 
(Art.  400),  and  finding  the  positive  roots  of  the  resulting 
equation,  we  may  obtain  the  negative  roots  of  the  proposed 
equation. 

Remark. — It  is  generally  easier  to  find  the  first  decimal  figure 
of  the  root  by  trial  than  by  Sturm's  theorem. 

433.  To  illustrate  this  method,  let  it  be  required  to  find 
the  positive  root  of  the  equation  o--^ — 4x — 10.*768649=0. 

We  readily  find  that  X  must  be  greater  than  5,  and  less  than  6j 


HORNER'S  METHOD  OF  APPROXIMATION.  389 

therefore,  a=^5.     We  then  proceed  to  transform  this  equation  into 
another  whose  roots  shall  be  less  by  5.     (See  Art.  410.) 


5) 

1-4 

—10.768649 

+5 

+  5 

+1 

—  5.768649 

+5 

+6 

1st  Trans,  eq.  . 

. 

2/2+6^ 

—  5.768649=0. 

Here  we  may  find  the  value  of  y  nearly,  by  dividing  5.7  by  6, 
which  gives  .9;  but  this  is  too  great,  because  we  neglected  y^.  If 
we  assume  2/=-8,  and  deduct  2/^=.64  from  5.7,  and  then  divide  by  6, 
we  see  that  y  must  be  .8,  Let  us  now  transform  the  equation  into 
another  whose  roots  shall  be  less  by  .8. 


.8) 

1 

+6 
.8 

—5.768649 
+5.44 

+6.8 
.8 

7.6 

—  .328649 

Trans,  eq.    . 

.    z^.-\-7.6z~ 

-.328649=0. 

The  approximate  value  of  z  in  this  equation  is  the  second  decimal 
figure  of  the  root.  This  is  readily  found  by  dividing  the  absolute 
term  by  the  coefficient  of  z,  the  first  term,  z^,  being  now  so  small 
that  it  may  be  neglected.     Thus,  .328-^7.6=.04=s. 

We  next  diminish  the  roots  of  the  last  equation  by  .04. 


s 

.04) 

1            +7.6 
.04 

+7.64 
.04 

-  328649 
.3056 

.023049 

3d  Trans,  eq.     . 

+7.68 
....  2:^2  1  7.68.:;' 

~.023049z 

390  RAY'S  ALGEBRA,  SECOND  BOOK. 

Here  ^  is  nearly  .023^7.68=:.003=r<. 

By  diminishing  the  roots  of  the  last  equation  by  .003,  we  have 

t 

.003)  1  +7.08  —.023049 

.003  .023049 


4-7.683  .0 

The  remainder  being  zero,  shows  that  we  have  obtained  the  exact 
root,  which  is  5.843. 

By  changing  the  sign  of  the  second  term  of  the  proposed  equation, 
we  have  a:^+4a: — 10.768649=0.  The  root  of  this  equation  may  be 
found  in  a  similar  manner;  it  is  1.843,  Hence,  the  two  roots  are 
+5.843  and  —1.843. 

Ex.  2. — To  illustrate  this  method  further,  let  us  form 
the  equation  whose  roots  are  3,  -f-|/2,  — 1/2,  which  gives 
ic' — Zt^ — 2a;-(~^=^-  ^^^  ^^  "^^  ^^  required  to  find,  by 
Horner's  method,  the  root  which  lies  between  1  and  2 ; 
that  is,  |/27 

One  root  lies  between  1  and  2;  hence,  «=1,  and  the  first  step  is 
to  transform  the  equation  so  as  to  diminish  its  roots  by  1. 

a 

1)        1        -3        -2        +6 

+1        _2        _4 

~Il2      "^4*    "+2 
-1        -5  T'     5 

0 

Hence,  ?/''rt0?/2_5y+2=0,  is  the  first  transformed  equation. 
By  dividing  the  absolute  term  2  by  5,  the  trial  divisor  or  coefficient 
of  ?/,  we  find  7*=:.4,  and  proceed  to  transform  the  equation  so  as  to 
diminish  its  roots  by  .4. 


HORNER'S  METHOD  OF  APPROXIMATION. 


391 


.4)        1 

zbO 
.4 

—5 
.16 

-4.84 
+   .32 

—4.52 

+2 
—1.936 

.4 
.4 

+  .064 

V''     .064 

.8 
.4 

^     T^'-~4.52 

=m 


1.2 


This    gives    z^-}-1.2z^—4.52z-\-Mi^0,    for   the  2d    transformed 
equation;  and  for  s,  the  next  figure  of  the  root,  .01. 

Transform  this  equation  so  as  to  diminish  its  roots  by  .01. 


.01) 


+1.2 
.01 

.01 

1^22 
.01 

T23 


-4.52 
.0121 


-4.5079 
.0122 

-4.4957 


-f.064 
—.045079 

+.018921 


^0189 
4:495' 


.004 


This   gives  ^'S+l. 232^— 4.49572;'+.01 8921:^0,  for  the  3d  trans- 
formed equation ;  and  for  the  next  figure  of  the  root  ^=.004. 
Transform  this  equation  so  as  to  diminish  its  roots  by  .004. 


t 

.004) 


+1.23 
.004 

1.234 
.004 

1.238 
.004 

1.242 


—4.4957 
+  .004936 

-4.490764 
+  .004952 

—4.485812 


+.018921 
—.017963056 

.000957944 


We  may  obtain  several  of  the  succeeding  figures  accurately  by 
division;  thus,  .000957944--4.485812==.0002135,  which  is  true  to 
the  last  decimal  place,  as  will  be  found  by  extracting  the  square 
root  of  2.     Hence,  a;:^l. 41 42135. 


392  RAY'S  ALGEBRA,  SECOND  BOOK. 

In  practice  it  is  customary  to  make  some  abridgments.  Thus, 
mark  with  a  *  the  coefficients  of  the  unknown  quantity  in  each 
transformed  equation  instead  of  rewriting  it.  Also,  when  the  root 
is  required  only  to  five  or  six  places  of  decimals,  use  about  this 
number  in  the  operation. 


3.  Given  a:*—Sx^-{-14x^-\~4:X—S=0,  to  find  a  value  of  x. 


5.236068; 


—8 
5 

OPERA 

+14 
-15 

-  1 
10 

9 
35 

*44 
2.44 

TION. 

-f4 
—5 

—1 
45 

*44 
9.288 

—8 
—5 

—3 
5 

*— 13 
10.6576 

2 
5 

*—  2.3424 

1.93880241 

7 
5 

53.288 
9.784 

*—    .40359759 
.39905490 

*12 

.2 

46.44 

2.48 

-63.072 
1.554747 

«_    .00454269 
.00400954 

12.2 
.2 

48.92 
2.52 

64.626747 
1.566321 

?i:_    .00053315 

12.4 
.2 

*51.44 
.3849 

-66.193068 
.31608 

12.6 

.2 

51.8249 
.3858 

66.50915 
.31656 

*12.8 
.03 

52.2107 
.3867 

«66.82571 

12.83 
.03 

*52.5974 
.08 

12.86 
.03 

52.68 
.08 

12.89 
.03 

52.76 

*12.92 


IIORNERS  METHOD  OF  APPROXIMATION.  3<J3 

As  the  root  is  found  only  to  six  decimal  places,  carry  the  true 
divisor  for  the  third  figure  (6)  to  five  decimal  places.  This  divisor 
is  66.50915,  which,  multiplied  by  .006,  gives  eight  decimal  places; 
and  the  dividend  ought  to  be  carried  thus  far,  to  make  the  figure  in 
the  sixth  decimal  place  of  the  root  correct. 

The  divisor,  66.825,  for  the  fifth  figure  of  the  root,  requires  to  be 
carried  only  to  three  decimal  places,  for  the  product  of  this  number 
by  .00006  gives  eight  decimal  places,  as  it  ought  to  do.  So  the 
divisor  for  the  last  figure  (8)  of  the  root  would  require  to  be  carried 
only  to  two  decimal  places. 

The  numbers  in  the  preceding  columns  require  to  be  carried  to 
still  fewer  places,  as  will  readily  be  perceived. 

The  last  three  figures  of  the  root  may  be  obtained  merely  by 
division;  thus,  .00454269--66.S2571  =.000068,  nearly. 

Observe  that  where  decimals  are  omitted,  we  always  take  the 
figure  next  to  the  omitted  places,  to  the  nearest  unit.  Thus,  .07752  is 
nearer  .08  than  .07;  therefore,  the  former  is  taken. 

434.  Horner's  method  may  be  applied  to  equations  of 
any  degree,  and  is  the  most  elegant  method  of  approxima- 
tion yet  discovered.    It  may  be  expressed  by  the  following 

Bule. — 1.  Fiiid,  hy  trials  or  hy  Sturm's  theorem,  the  in- 
tegral part  of  the  requij-ed  root. 

2.  Transform  the  equation  (Art.  410)  into  another  whose 
roots  shall  he  those  of  the  proposed  equation,  diminished  hy 
the  part  of  the  root  already  found. 

3.  With  the  absolute  term  in  the  first  transformed  equation 
for  a  dividend,  and  the  coefficient  of  x  for  a  divisor,  find 
the  first  decimal  figure  of  the  root. 

4.  Transform  the  last  equation  into  another  whose  roots 
shall  he  diminished  hy  the  part  of  the  root  already  found, 
and  from  the  first  tico  terms  of  this  equation,  find  the  second 
figure  of  the  root. 

5.  Continue  this  process  till  the  root  is  found  to  the  required 
degree  of  accuracy. 

6.  To  find  the  negative  roots,  change  the  signs  of  the  alter- 
nate terms,  and  proceed  as  for  a  positive  root. 


394  RAY'S  ALGEBRA,  SECOND  BOOK. 

Remarks. — 1.  If  any  figure,  found  by  trial,  is  too  great  or  too 
small,  it  will  be  made  manifest  in  the  next  transformation.  (See 
Art.  431.) 

2.  After  finding  three  figures  of  the  root,  the  next  three  may  gen- 
erally be  obtained  by  dividing  the  absolute  teft'm  by  the  coefficient 
of  a:. 

Find  at  least  one  value  of  x  in  each  of  the  following : 

1.  a:^-]-5a:— 12.24=0 Ans.  a:=1.8. 

2.  x^-f  12a;— 35.4025=0 Ans.  a:=2.45. 

3.  4^2— 28a;— 61.25=0 Ans.  a;=8.75. 

4.  Sa;'^— 120a;+394.875=0.       .     .   Ans.  a;=10.125. 

5.  5a;^— 7.4a;-16.08=0 Ans.  a;=2.68. 

6.  .T2_6a;+6=0 Ans.  a;=4.73205. 

7.  a:«4-4a;2— 9a;— 57.623025=0.    .      .  Ans.  a:=3.45. 

8.  2a;3— 50x4-32.994306=0.    .     .     .  Ans.  a;=4.63. 

9.  a;»-f  4a;'— 5a;— 20=0.     .     .     .     Ans.  a;=2.23608. 

10.  a;'— 2a;— 5=0 Ans.  a:=2.0945515. 

11.  a;'^-flO.T2— 24a;— 240=0.         Ans.  a;=4.8989795. 

12.  a;*— 8a;3+20a;2— 15a;+.5=0.     Ans.  a;=1.284Y24. 

13.  a;*-59a:-^4- 840=0.  .     .     .       Ans.  a;=4.8989795. 

14.  2a;*-f  5a;»-h4rc'*+3a;=8002.       Ans.  a;=7. 335554. 

15.  a;5-[-2a;*+3:c3+4x'^+5a;=54321.  A.  a:=8.414455. 

433.  Te  extract  the  roots  of  numbers  hy  Horner^ s  Method. 

This  is  only  a  particular  case  of  the  solution  of  the 
equation  x^^N,  or  as"— N=0  ;  an  equation  of  the  n^^'  de- 
gree, in  which  all  the  terms  are  wanting  except  the  first 
and  last. 

In  performing  the  operation,  observe  that  the  successive  integral 
figures  have  the  same  relation  to  each  other  that  the  successive 
decimal  places  have  in  the  previous  examples. 

In  extracting  any  root,  point  off  the  given  number  into  periods, 
as  in  the  operation  by  the  common  rule. 


APPROXIMATION  BY  DOUBLE  POSITION.  395 

For  an  example,  let  it  be  required  to  find  the  cube 
root  of  129778*75  ;  that  is,  one  root  of  the  equation 
o;^— 12977875=0- 

235)  1 


0 

2 

0 
4 

12977875 

8. 

2 
2 

4 

8 

4977 
4167 

4 

2 

*12 
189 

810875 
810875 

-6 

1389 

3 

198 

G3 

-1587 

3 

3475 

G6 
3 

162175 

*69 

5 

695 

The  reason  for  placing  the  figures  as  they  are  in  the  successive 
columns,  will  be  readily  understood  by  using  the  numbers  200  and 
30,  instead  of  2  and  3. 

By  the  same  method  find 

2.  The  cube' root  of  34012224.     .     .     .     Ans.  324. 

3.  The  cube  root  of  9 Ans.  2.080084. 

4.  The  cube  root  of  30 Ans.  3.107233.      . 

5.  The  fifth  root  of  68641485507.   .     .     Ans.  147. 


APPROXIMATION    BY    DOUBLE    POSITION. 

4S6.  Double  Position  furnishes  one  of  the  most  use- 
ful methods  of  approximating  to  the  roots  of  equations. 
It  has  the  advantage  of  being  applicable,  whether  the  equa- 
tion is  fractional^  radical,  or  exponential,  or  to  any  other 
form  of  function. 


396  RAY  S  ALGEBRA,  SECOND  BOOK. 

Let  X=:0,  represent  any  equation ;  and  suppose  that  a 
and  Z>,  substituted  for  cc,  give  results,  the  one  too  small, 
and  the  other  too  great,  so  that  one  root  lies  between  a 
and  h.     (Art.  403.) 

Let  A  and  B  be  the  results  arising  from  the  substitu- 
tion of  a  and  b  for  x,  in  the  equation  X=;0.  Let  x=a-{-h, 
and  b=a-^Jc]  then,  if  we  substitute  a-}-h  and  a-{-7c  for  x, 
in  the  equation  X=:0,  we  shall  have 

X=.A-f  A'A-f-JA^'A^-f ,  etc. 
B==A+A7^+iA"A;'^+,  etc. 

Here,  A^,  A^^,  etc.,  are  the  derived  functions  of  A  (Art.  411). 
Now,  if  h  and  k  be  so  small  that  their  second  and  higher  powers 
may  be  neglected  without  much  error,  we  shall  have 


X-A=:--A^/i  nearly; 

B— A=A^/^       "      . 

Whence, 

B-A 

X— A::  A'Jc:  A'h  :  7c:  %; 

Or,    .     . 

B— A 

h         :  :  X— A  :  /i,  (Art.  270) 

Or,     .     . 

B-A 

b—a  :  :  X— A  :  h,  since  k= 

a. 

Hence,  we  have  the  following 

Knle. — Find,  by  trial,  two  numbers  which,  substituted 
for  X,  give  one  a  result  too  small,  and  the  other  too  great. 
Then  say,  As  the  difference  of  the  results  is  to  the  difference 
of  the  suppositions,  so  is  the  difference  between  the  true  and 
the  first  result,  to  the  correction  to  be  added  to  the  first  sup>- 
position. 

Substitute  this  approximate  value  for  the  unknown  quan- 
tity, and  find  whether  it  is  too  small  or  too  great;  then, 
take  two  less  numbers,  such  that  the  true  root  may  lie 
between  them,  and  proceed  as  before,  and  so  on. 

It  is  generally  best  to  begin  witli  two  integers  which  differ  from 
each  other  by  unity,  and  to  carry  the  first  approximation  only  to  one 
place  of  decimals.  In  the  next  operation  make  the  ditference  of  the 
suppositions  0.1,  and  carry  the  2d  quotient  to  two  places,  and  so  on. 


NEWTON  S  METHOD  OF  APPROXIMATION.  397 

1.  Given  x^-{-x''-\~x=100,  to  find  x. 

Here,  X  lies  between  4  and  5.     Substitute  these  two  numbers 
for  X  in  the  given  equation,  and  the  result  is  as  follows : 


4    .     . 

.     .     .     X     .     .     . 

.     .     .    5 

64    .     .     . 

.    .    .    x^  .    .    . 

.    .    .    125 

16    .     .    . 

.    .    .    a;2  .    .     . 

.    .    .      25 

4    .     .    . 

.     .     .     X     .     .     . 

.    .    .        5 

84    .     .    . 

.     .     results     .     . 

.    .     .    155 

55     .     .     . 

.    .     .    5     .    .    . 

.    .     .    100 

84 4 84 

71  :  "1  :  :  ~Tq     :  0.22, 

therefore,  a:=4.2,  the  first  approximation. 

Substituting  4.2  and  4.3  for  x,  and  proceeding  as  before,  we  get 
for  a  second  approximation  x=4.264:.  Assuming  a:=4.264  and 
4.265,  and  continuing,  we  obtain  a:=4.2644299,  nearly. 

Find  one  root  of  each  of  the  following  equations : 
2.  x'-\-S0x=.420 Ans.  x=6.1'J010S. 


3.  144x3— 973a^=:319. 

4.  x'-^10x-'-\-bx=2600. 

5.  2x'-ir^x'—4x=10.     . 

6.  x'—x''-\-2x'-\-x=4.     . 


.  .  Ans.  .T=2.75. 
Ans.  a-=ll. 00679. 
.  Ans.  a:=l. 62482. 
.  Ans.  a:=rl.l4699. 


7.  f1x'-^4x'JrVl0x{2x—l)=-2S.   A.  a;^4.51066. 

437.  Newton's  Method  of  Approximation. — Thio 
method,  now  but  little  used,  is  briefly  as  follows : 

Find,  by  trial,  a  quantity  a  within  less  than  0.1  of  the 
value  of  the  root.  Substitute  a-j-i/  for  x  in  the  given  equa- 
tion, and  it  will  be  of  this  form 

A+AV+iAV+iA"y+,  etc.,  :=0     (Art.  411), 
where  A,  A',  A",  etc.,  are  what  the  proposed  equation,  the 
first  derived  polynomial,  etc.,  become  w^hen  x=a. 


398  RAY'S  ALGEBRA,  SECOND  BOOK. 

From  this  equation,  by  transposing  and  dividing, 

A         A^''  K^^^ 

We  find  ?/=:— ^/_i  — 2/2— ^— 2/"*— ,  etc.;  and  since 

y  is  <0.1,    ?/2  will  be  <0.01,   2/'''<0.001,  and  so  on. 

Therefore,  if  the  sum  of  the  terms  containing  2/^,  2/^  etc.,  be  less 
than  .01,  we  shall,  in  neglecting  them,  obtain  a  value  of  y  within 

.01  of  the  truth.     Putting  y=—~^  we  havea:=a— ^.      This  will 

diflFer  from  the  true  value  of  x  by  less  than  .01. 

Now,  put  b  for  this  approximate  value  of  x,  and  let  X::^b-\-Z',  we 
have  then  as  before 

B-f  B^2;+^B^^2:2_^^B^^^03_[_^  etc.,  =^0; 

and  as  z  is  supposed  to  be  less  than  .01,  z'^  will  be  <.0001.  If^ 
then,  we  neglect  the  terms  containing  2;2,  z^,  etc.,  we  shall  obtain  a 
probable  value  of  Z  within  .0001 ;  and  so  on.  Applying  the  succes- 
sive corrections,  we  obtain  the  value  of  x. 


Newton  gave  but  a  single  example,  viz. 

Required  to  find  the  value  of  x  in  the  equation  x^ — 2.t 
5=0.  Ans.  0^=2.09455149. 


CARDAN'S  RULE  FOR  SOLVING  CUBIC  EQUATIONS. 

43S.  In  its  most  general  form,  a  cubic  equation  may- 
be represented  by 

x?-\-2yx'^-\-qx-\-r=^0 ; 

but  as  we  can  always  take  away  the  second  term,  (Art.  407,) 
we  will  suppose,  to  avoid  fractions,  that  it  is  reduced  to 
tiie  form 

a:»_|_3^a;-f2r=0. 

Assume  x=zy-{z^  and  the  equation  becomes  , 

y3_^2;3_|_3y2;(2/-f2:)+3(7(2/-f2)4-2r=0. 

Now,  since  we  have  two  unknown  quantities  in  this  equation,  and 
have  made  only  one  supposition  respecting  them,  we  are  at  liberty 


CARDAN  S  SOLUTION  OF  CUBIC  EQUATIONS.        399 

lo  make  another.     Let^  therefore,  yz:=^ — g.     Substituting,  we  have 
^•'5_|_2:3_[_2r=0;  but  since  yz=—q,  z^'z 


y. 


q 


Hence, 2/3— ^-|-2r^0; 

Whenee, 2/'3= -r-fy'r^ip^. 

And  similarly,      ,     .     z^^^ — r — \'"f'^'\-(j^', 

the  radical  being  positive  in  one,  and  negative  in  the  other,  by 
reason  of  the  relation  yz-= — q. 
And  since  a;=2/-|-2;,  we  have 


This  formula  will  give  one  of  the  roots.  The  others  may  be  found 
by  reducing  the  equation  (Art.  396,  Cor.  1). 

-f-  439.  If  r^-fgs  be  negative,  that  is,  if  r'+$''<0,  the 
values  of  x  become  apparently  imaginary  when  they  are 
actually  real,  and  we  shall  now  show  that 

Cardan's  Method  of  Solution  does  not  extend  to  tJiose  cases 
in  which  the  equation  has  three  real  and  unequal  roots. 

Suppose  the  one  real  root  (Art.  401,  Cor.  3),  to  be  a;  and  the 
other  two  arising  from  the  solution  of  a  quadratic  to  be  6-|-y/3c, 
und  b—^/Sc,  in  which,  if  3c  be  positive,  the  roots  are  real,  and  if  3c 
be  negative,  they  are  imaginary ;  and  because  the  second  term  of 
the  equation  is  0,  we  have  (Art. 


0=a+(6+ |/3c)  +  (6— i/3c)=ra+26 ; 
3g=aX26+^^— 3c=— 362— 3c ; 
2r=—a{b^—3c):=2b^~Qbc. 

Hence,  we  have 

r^-\-q^={b^~3bcy—{b^-^e)^=—9b^e-\-eb-e^—c^ 
=._c(362_.c)2  ...  ^/r2+g3^(362_c)^irc. 

Now,  this  expression  is  real  when  C  is  negative,  and  imaginary 
when  c  is  positive,  or  when  the  equation  has  three  real  roots. 


400  RAY'S  ALGEBRA,  SECOND  BOOK. 

If  c=0,  the  roots  are  a,  6,  and  6;  hence,  Cardan's  Rule  is  appli- 
cable to  equations  containing  two  equal  roots. 

440.  In  illustration  of  the  apparent  paradox  that  when 
the  roots  of  the  quadratic  equation,  Art.  438,  are  imagin- 
ary, the  roots  of  the  cubic  equation  are  all  real,  take  the 
following 

Example. — To  find    the    three   roots  of  the   equation 
By  substituting  y-\-z  for  x,  we  have 

And,  since     .     .     .    3i/z=15,   y^-{-z^—4=0. 

From  the  solution  of  these  equations,  we  obtain  y^=2-\-lly'^—i. 
By  actual  multiplication,  we  find  that  p=2-{-y^ — 1  ;  likewise, 
s^>rz=2— 11/^,  and  z=2—^^^. 

Hence,  x=y-^z={2i-y^-[)-^{2—^^^)=4. 

By  dividing  the  given  equation  by  X — 4,  we  find  the  other  two 
roots  are  x=—2-\-y'3,  and  —2 — ^3. 

As  no  means  have  yet  been  discovered  for  reducing  the 
imaginary  forms  to  real  values.  Cardan's  rule  fails  when 
all  the  roots  are  real.  This  is  the  Irreducible  Case  of  cubic 
equations. 

441.  The  following  examples,  containing  one  real  and 
two  imaginary  roots,  may  be  solved  b'y  Cardan's  rule. 

When  the  equation  contains  the  second  term,  remove 
it  (Art.  40 Y),  and  reduce  the  equation  to  the  form 
x'-^Sqx-\-2r=0. 

Then,  x^f{—r-\-^r^^^)-{-f{—r—^/'f2-j-q^),  will  be  the  real 
root  of  the  proposed  equation. 

Having  the  real  root,  the  imaginary  roots  may  be  found  by  reduc- 
ing the  equation  to  a  quadratic  (Art.  396,  Cor.  1). 


RECIPROCAL  OR  RECURRING  EQUATIONS.  401 

1.  Solve  the  equation  v^-j-Sv''^9v—lS=0. 

Substituting  x — 1  for  v  (Art.  407),  we  have  x^-\-6x — 20=0. 
Comparing  this  with  the  equation  x^-\-Sqx-]-2r^:zzO,  we  find  q=2, 
7*:^— 10 ;  hence, 

a;=:f  (1 0+ y'lOS)  4- f  (10— i/108j=2.732— .732=2. 
Whence,  ^;=a>— 1=2— 1=1. 
The  other  two  roots  are  easily  found  to  be  —  IdtSy' — 1. 

2.  0)3— 9a:+28..:=0.  .     .     .     Ans.  x=—4,  2±^=8. 

3.  x'-{-6x—2=0.     .     .  Ans.  a:=/4— /2=.32748. 

4.  x'—Gx'Jr'^Sx—10=0.         Ans.  x=2,  2±y/3i. 

5.  a^—9x'^6x—2=.0 Ans.  x=SM66l4. 

Remark  . — Cardan's  Rule,  together  with  those  of  Ferrari,  Euler, 
Descartes,  and  others,  are  regarded,  since  the  discovery  of  Horner's 
method,  and  Sturm's  theorem,  as  little  more  than  analytical  curi- 
osities. 

RECIPROCAL  OR  RECURRING  EQUATIONS. 

442.  A  Recurring  or  Reciprocal  Equation  is  one  such 
that  if  a  be  one  of  its  roots,  the  reciprocal  of  a  will  be 
another. 

Proposition  I. — In  a  recurring  equation  the  coefficients^ 
when  taken  in  a  direct  and  in  an  inverse  order ^  are  the  same. 

Let  a;^+Aa;"-i+BiC"-2 ^Sa:2-f  Ta:+V=0,  ^e  a  recurring 

equation;  that  is,  one  that  is  satisfied  by  the  substitution  of  the  re- 
ciprocal of  X  for  X.     This  gives 

and  multiplying  by  X'*, 

1+Aa:+Ca;2 -f-Sa;''-2+Ta;«-i+Va:''=0, 

which  proves  the  proposition. 
2d  Bk.  84 


402  RAY'S  ALGEBRA,  SECOND  BOOK. 

Such  equations  are  called  Recurring  Equations,  from  the  forms  of 
iheir  coe^cienls ;  and  Reciprocal  Equations,  from  the  forms  of  their 
roots. 

Proposition  II. — A  recurring  equation  of  an  odd  degree^ 
has  one  of  its  roots  equal  to  -}-l,  when  the  signs  of  the  like 
coefficients  are  different^  hnt  equal  to  — 1,  when  their  signs 
are  alike. 

Since  every  power  of  -fl  is  positive;  when  the  signs  of  the  like 
coefficients  are  different,  if  we  substitute  -f  1  for  x  the  corresponding 
terms  will  destroy  each  other. 

AVhen  the  signs  of  the  like  coefficients  are  the  same,  since  one  will 
belong  to  an  odd,  and  the  other  to  an  even  power,  if  we  substitute 
— 1  for  X,  the  corresponding  terms  will  destroy  each  other. 

Such  equations  may,  therefore,  be  redmced  one  degree  lower  by 
dividing  by  x — 1,  or  x-]-!. 

Proposition  III. — A  recurring  equation  of  an  even  degree^ 
ivhose  like  coefficients  have  opposite  signs,  is  divisible  by  x^ — 1, 
and  tJierefore  two  of  its  roots  are  -j-l,  and  — 1. 

Leta:2'*-fAa;2"-i-|-Ba:2"-2 _Ba;2— Aa;— 1=0,  be  an  equa- 
tion of  the  kind  specified.  It  may  evidently  be  arranged  thus,  and 
be  divisible  by  x^~l     (Art.  83). 

(a:2n_|)_^Aa;(a:2»-2_l)_>.Ba;2(a:2"-4_l)-|-,  etc.  .  .  .=0. 

Corollary. — An  equation  of  this  form  may  therefore  be 
reduced  two  degrees  lower  by  either  common  or  synthetic 
division. 


Proposition  IV. — Every  recurring  equation  of  an  even 
degree  above  the  second^  may  he  reduced  to  an  equation  of 
half  that  degree,  when  the  signs  of  Hie  correspoTuiing  tenns  are 
alike. 

For,  a;2n_Aa;2»-i_^Bz2»-2_,  ete 4-Bx2— Az-f  1=0,  by 


RECURRING  EQUATIONS.  403 

dividing  by  a:",  and  collecting  the  pairs  of  terms  equi-distant  from 
the  extremes,  becomes  of  the  form 

(  ^"+^'  )  -^  (  ^"-  +^  )  +"  (  ^""+x-=-^  )  ->  ^"=-  =°- 
Let  a:-j — —2;    then,  x^-\ — 2=^^ — 2,  by  squaring;  also, 

and  generally  (  x-'+i  ^  =  (  a;-i  +  ^  )z-(  ^"-'+^  )• 

Hence,  each  of  the  binomials  may  be  expressed  in  terms  of  z,  and 
the  resulting  equation  will  be  of  the  n''*  degree. 


1.  Given  x* — bx^-\-6x'^ — bx-\-l=:^0,  to  find  x. 
Here,     ....    a:2-5rc-f  6— -  +  ^  =  0, 

Let  X-] — —z;  then,  z^ — 50+4=iO,  and  z=4,  or  1 ; 

Putting  x-\ —  equal  to  each  of  these  values  of  2,  we  obtain  for  the 

four  values  of  X,  2=bi/3~and  ^(1±t/— 3). 

The  second  of  tliese  values  is  the  reciprocal  of  the  first,  and  the 
fourth  of  the  third,  as  may  be  shown,  thus : 

1  1  2-T./3      2-/3 

2+v^'"2+v/"3^2-v/3~  4-3  =^2-i/3. 


EXAMPLES  IN  RECURRING  EQUATIONS. 

1.  a^*—10.7:3-^26.T2— 10^:4-1=0. 

Ans.  x=S^2y'%  2±|/3. 

2.  x*—^x^-^2x'—.^x-j-l=0. 

Ans.  x=2,    1,    ±1/^. 


404  RAY  S  ALGEBRA,  SECOND  BOOK. 

3.  x'~Sx'-^Sx—l=0.  Ans.  x=^l,    l(3±|/5). 

4.  x'-~llx'-\-11x'-{-11x''—llx-\-l=0. 

Ans  x~     1    ^±1^    ^-V^^'    ^+l/^     3-T/^ 

5.  4ic«— 24a:^+57a^*— 73a:34-57a;2-.24x-h4=r0. 

Ans   x-^^    ^    2    ^    1±1^?    l-l/^ 


BINOMIAL    EQUATIONS. 

443.  Binomial  equations  are  those  of  the  form 

3/"±A:=0. 

Let *[/A=a;  thatis,  A=a"; 

Then, 2/"±a"=0. 

Let     .     .     ,      y=ax',  then,  a"a;"dra"=:0. 
Or, a:"=tl=0, 

which  is  a  recurring  equation. 

444. — I.  The  roots  of  the  equation  x"=tl=:0,  are  all 
unequal ;  for  the  first  derived  polynomial  nx"~^^  evidently 
has  no  divisor  in  common  with  ic"±l,  and  therefore  there 
are  no  equal  roots  (Art.  414). 

II. — If  n  be  even,  the  equation  a;" — 1=0,  or  x"=l,  has 
two  real  roots,  -|-1  and  — 1,  and  no  more,  because  no  other 
real  number  can,  by  its  involution,  produce  1. 

By  dividing  x"— 1=0  by  (x-\-l)(x—l)—x'^—l,  we  have 
a  recurring  equation,  having  n — 2  imaginary  roots. 


BINOMIAL  EQUATIONS.  405 

For  example,  the  equation  x^=l,  or  x^ — 1=0  divided 
by  ic* — 1  gives  x*-\-x^-^l:=0;  whence, 

,  /(-i±j/=ri 
"=*Vl — 2^  I- 


This  gives  for  the  six  roots  of  1 

+1,  -1, 

III. — If  n  be  odd,  the  equation  x^ — 1=0  has  only  one 
real  root,  viz.:  -fl ;  for  -|-1  is  the  only  real  number  of 
which  the  odd  powers  are  -\-l. 

Dividing  x^ — 1=0  by  x — 1,  we  have 

a  recurring  equation,  having  n — 1  imaginary  roots. 

For  example,  the  equation  x^=l,  or  x^ — 1=0,  divided 
by  X — 1,  gives  a:"^-j-a:-f-l=0;  whence,  a;= ^ . 

Hence,  the  three  third  roots  of  1  are 


1  -1+1/-3     -1-^-3 

^'  2         '  2         • 


IV. — If  n  be  even,  the  equation  .t"-}-1=0,  or  ic"= — 1, 
has  no  real  root,  since  ^ — 1  is  then  impossible.  Hence, 
all  the  roots  of  this  equation  are  imaginary. 


406  RAY'S  ALGEBRA,  SECOND  BOOK. 

For  example,  the  four  roots   of  the   recurring  equation 
ic*-f  1=0  (Art.  442),  are 

-^14-1/^=1       —l-y^l       l+v/=i       l~y—l 

1/2       '  ^/2       '         -/2      '         1/2"     ' 


V. — If  n  be  odd,  the  equation  x"'-{-l=0,  or  x^= — 1,  has 
one  real  root,  viz.:  — 1,  and  no  more,  because  this  is  the 
only  real  number  of  which  an  odd  power  is  — 1. 

For  example,  a:^-j-^l=0,  divided  by  x-^1,  gives  x^ — x 
-j-l=0  ;  whence,  x= ^ . 

Therefore,    the    three    third    roots    of   — 1,    are    — 1, 
o 5  ana ^ . 


Binomial  equations  have  other  properties,  but  some  of 
them  can  not  be  discussed  without  a  knowledge  of  Ana- 
lytical Trigonometry. 

1.  Find  the  four  fourth  roots  of  unity. 

Ans.  +1,  -1,  +i/— 1,  — i/— 1. 

2.  Find  the  five  fifth  roots  of  unity. 

Ans.  1, 
i{v/5-l  +  y(-10-2|/5)}, 
i{l/5-l-|/(-10-2v/5)}, 

--i{l/5+l+l/(-10+2i/  5)1, 
~|{l/5+l-|/(-10+2v^5)|. 


THE    END, 


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